CORAL GABLES, Fla.-In their first one-on-one debate last night, President Bush and Senator Kerry traded sharp words over the war in Iraq.

Mr. Kerry repeatedly denounced the invasion, describing it as a reckless diversion from the broader war on terror.

“This president has made, I regret to say, a colossal error of judgment. And judgment is what we look for in the president of the United States of America,” Mr. Kerry said.

Mr. Bush painted the war as an essential part of the global terror fight.

“In Iraq, we saw a threat, and we realized that after September the 11th, we must take threats seriously, before they fully materialize. Saddam Hussein now sits in a prison cell,” the president said. “America and the world are safer for it.”

Mr. Bush also charged that Mr. Kerry’s criticism of the war made him unfit to be commander in chief. He pointed in particular to the senator’s description of the war in Iraq as “the wrong war in the wrong place at the wrong time.”

“I don’t see how you can lead this country to succeed in Iraq if you say wrong war, wrong time, wrong place,” Mr. Bush said. “What message does that send our troops? What message does that send to our allies? What message does that send the Iraqis?”

Mr. Bush also said Mr. Kerry’s vote against an $87 billion funding bill for the war and his awkward explanation of the vote showed a vacillation that could be dangerous if the senator becomes president.

“Not what a commander in chief does when you’re trying to lead troops,” the president said.

Mr. Kerry had clearly prepared for such a line of attack.

“When I talked about the $87 billion, I made a mistake in how I talk about the war. But the president made a mistake in invading Iraq,” Mr. Kerry said. “Which is worse?”

Mr. Kerry also accused Mr. Bush of abandoning efforts at the United Nations in 2002 to disarm Iraq peacefully through weapons inspections.

The president said Mr. Kerry’s endorsement of that approach suggested that he had not learned the lessons of the September 11, 2001 terrorist attacks.

“That’s kind of a pre-September 10th mentality to hope that resolutions and inspections would make this world a safer place,” Mr. Bush said.

At times, Mr. Kerry seemed to exasperate the incumbent president. Mr. Bush grimaced in response to some of his opponent’s answers. He appeared a bit defensive in the first half hour of the debate, but seemed more comfortable after that.

After the senator said several times that Mr. Bush had diverted resources that should have been used to hunt down Osama bin Laden, the president shot back, “Of course, I know Osama Bin Laden attacked us. I know that.”

Both candidates also offered some answers that must have made their staffers cringe.

In response to one question, Mr. Kerry appeared to say that the American president should only take pre-emptive action if the world agrees.

“You’ve got to do it in a way that passes the test, that passes the global test,” the senator said.

Mr. Bush jumped on the comment.

“I’m not exactly sure what you mean,” the president said. “I just think trying to be popular, kind of, in the global sense, if it’s not in our best interest, makes no sense. I’m not going to make decisions that are wrong for America.”

At one point, Mr. Kerry lambasted the president for not doing enough to strengthen terror defenses at home. “This president thought it was more important to give the wealthiest people in America a tax cut before we invest in homeland security,” the senator charged.

In response, Mr. Bush seemed to say that the programs Mr. Kerry was proposing were too expensive.

“I don’t think we want to get to how he’s going to pay for all these promises. It’s like a huge tax gap,” the president.

Both men agreed that the gravest threat facing America is the prospect of nuclear weapons falling into the hands of terrorists. However, Mr. Kerry accused Mr. Bush of doing little to prevent the spread of such weapons on his watch.

“The president had four years to do something about it. North Korea’s got more weapons. Iran’s moving toward weapons,” the senator said.

During the course of the debate, Mr. Kerry made at least five references to his service as a Navy officer in Vietnam. He also used the word “kill” repeatedly as he discussed his plans to deal with Osama bin Laden and other terrorists.

On at least four occasions, Mr. Bush underscored the need for America to take the fight to the terrorists on foreign soil.

“We have a duty to defeat this enemy,” the president said.

“The best way to defeat them is to never waver, to be strong, to use every asset at our disposal, is to constantly stay on the offensive and, at the same time, spread liberty.”

The candidates also tangled about American policy towards Iran. Mr. Kerry said he would offer Iran nuclear fuel if the country agreed not to use it for weapons.

“I think the United States should have offered the opportunity to provide the nuclear fuel, test them, see whether or not they were actually looking for it for peaceful purposes,” the senator said. “If they weren’t willing to work a deal, then we could have put sanctions together. The president did nothing.”

Mr. Bush suggested that Mr. Kerry had his facts mixed up.

“We’ve already sanctioned Iran. We can’t sanction them anymore. There are sanctions in place on Iran,” Mr. Bush said.

President Carter imposed a broad embargo on Iran after the 1979 hostage-taking at the American embassy there. Reagan re-imposed the embargo in 1987, and sanctions were strengthened during the Clinton administration. However, France, Germany, and other countries have continued to trade with Iran.

In his closing statement, Mr. Kerry turned back to Iraq. He stressed that he was not suggesting that America should cut and run there.

“I’m not talking about leaving. I’m talking about winning,” the senator said. “We need a fresh start, a new credibility, a president who can bring allies to our side.”

In his closing, Mr. Bush asked voters to elect him to another term.

“We’ve done a lot of hard work together over the last three and a half years,” the president said. “We’ve climbed the mighty mountain. I see the valley below and it’s a valley of peace.”

The two campaigns agreed to an unusually restrictive set of rules for this debate, as well as two more presidential debates that are to follow. The agreement required that each candidate remain behind his podium and precluded the use of any props. It also called for the installation of timing lights that would signal when a candidate’s time had expired.

Mr. Kerry’s campaign yesterday objected to the placement of these lights on the lecterns, saying that they would be distracting. Aides to Mr. Bush accused the Kerry camp of reneging on the agreement. The debate commission declined to remove the signal lights.

Both candidates appeared to comply strictly with the time limits, cutting short their remarks whenever the red light came on. The television networks, on the other hand, ignored the rules that the candidates sought to impose barring reaction shots. At several points, Mr. Bush was shown looking quizzically at Mr. Kerry as he spoke.

Several instant polls taken just after the televised contest ended found more voters saying that Mr. Kerry won the debate. In a CBS News poll, 44% of viewers said Mr. Kerry won, while 26% said Mr. Bush won. The debate was judged a tie by 30% of viewers, CBS said.

Mr. Kerry came into the debate as a clear underdog in the race. A Los Angeles Times survey released Wednesday found Mr. Bush with 51% support from likely voters nationwide. In the poll, Mr. Kerry had 45%, while independent candidate Ralph Nader had 2%. Other polls have shown Mr. Kerry trailing by a similar margin.

After the debate concluded, supporters of Mr. Kerry claimed victory. Some of Mr. Bush’s partisans said simply that he had held his own. A few of the president’s backers were a bit glum faced as they met reporters in the so-called spin room.

One prominent political analyst, Norman Ornstein of the American Enterprise Institute, said the debate was likely to deliver a boost to Mr. Kerry’s campaign.

“I think it’s a contest now. I think Kerry’s back in the game,” Mr. Ornstein said. He said he thought viewers were likely to see Mr. Kerry as, at a minimum, more competent than the Bush campaign has suggested in its withering campaign ads.

“People can see him as a president and that’s what he needed,” Mr. Ornstein said. He also predicted that in the coming days press accounts will talk of a Kerry comeback. “You’ve got a lot of reporters ready to write the ‘Rocky’ story,” the analyst said.

The selection of Miami for the debate venue reflected Florida’s status as a crucial swing state. However, a series of hurricanes that swept through the state over the past six weeks have left many residents with more immediate concerns than presidential politics. As of last evening, more than 275,000 Florida households remained without power, according to the state’s electric utilities.

Mr. Bush spent part of yesterday comforting hurricane victims at a relief center in Stuart, Fla., about 100 miles north of Miami. Mr. Kerry spent the day at a nearby hotel.

Security at the debate site was intense. Hundreds of police officers from around the area patrolled the university campus just south of Miami.

Backers of Mr. Bush and Mr. Kerry gathered on the highway just outside the campus. Cuban-American supporters of the president toted signs that said, “Viva Bush!”

In a yard nearby, a Kerry partisan erected a sign that disparaged Mr. Bush. “Bush is more destructive to America than all the hurricanes combined,” it said.