When the director and choreographer Josh Rhodes first staged “Spamalot” at Washington’s Kennedy Center last spring, he had big shoes to fill. A musical adaptation of the beloved film “Monty Python and the Holy Grail” by an original member of that renowned troupe, Eric Idle, had its Broadway premiere in 2005 under the direction of another icon, Mike Nichols; both Nichols and the show — which featured famously astute comedic actors such as David Hyde Pierce, Tim Curry, and Hank Azaria — scored Tony Awards.

Yet Mr. Rhodes obviously had a hunch that the world needed a good laugh earlier this year at least as much as it had back then. Six months after his production’s opening in the nation’s capital, it could well be argued that we need it even more, and this “Spamalot” delivers laughs by the bushel.

While the cast Mr. Rhodes has recruited for Broadway may not boast as many widely familiar names as the original did, it’s brimming with musical comedy favorites: James Monroe Iglehart, Michael Urie, and Leslie Rodriguez Kritzer reprise their Washington performances, as do veteran stage and screen actor Jimmy Smagula and a recent “Hamilton” star, Nik Walker. They’re joined at New York by two other popular troupers, Christopher Fitzgerald and Ethan Slater, as well as a “Saturday Night Live” alum, Taran Killam. 

In keeping with the film that inspired it, and the sketch-comedy tradition that Monty Python championed, “Spamalot” requires most of these players to juggle multiple roles. Mr. Iglehart, who first captured theatergoers’ hearts as a delightfully manic genie in the original Broadway production of “Aladdin,” stays in character as King Arthur, and is the closest thing to a straight man here; his royal authority may be served with a big dose of cheek, but it anchors the staging, freeing the others to go nuts.

That they do, with a blissful adroitness that stays true to the spirit of the source material. All of the principals demonstrate an affinity for the distinctly British brand of humor, at once carefully clever and unabashedly silly, that Mr. Idle and his colleagues practiced. There’s Mr. Urie, hilariously fey as Sir Robin, who wishes to be a knight but would rather dance than fight; and Mr. Slater, stumbling with purpose through the role of Not Dead Fred, who suffers inspired, Python-esque abuse at the hands and mouths of Robin and Mr. Killam’s Sir Lancelot.

Not surprisingly, perhaps, Mr. Killam — having only his second turn on a Broadway stage, following a stint several years ago as King George in “Hamilton” — invests particular skill and zest in his characters, who also include the more flamboyant “French Taunter.” For those unfamiliar with either “Spamalot” or “The Holy Grail,” suffice it to say the role demands an impeccably controlled comic lunacy, and Mr. Killam pulls the sword out of the stone. 

Ms. Rodriguez Kritzer, like Mr. Iglehart, is tasked with only one part, but it’s a doozy: The Lady of the Lake, whom Mr. Idle and collaborating composer John Du Prez created specifically for the stage musical, is a caricature of a narcissistic, if essentially well-meaning, diva. The actress dives in with a vengeance, gleefully chewing the scenery and delivering satirical numbers such as “The Song That Goes Like This” and “Diva’s Lament,” with the requisite vocal histrionics.

Jen Caprio’s costumes and Paul Tate dePoo III’s scenic and projection design can wink stylishly or be lusciously garish. In one of a few tweaks nodding to current pop and political culture, Congressman George Santos’s name is among those flashed above the stage, during the uproarious production number “You Won’t Succeed on Broadway”; the lyrics include, “Your show’s as good as dead/If you don’t have any Jews,” and several variations on that statement.

Some might argue that now is not an ideal time to joke about the prominence of Jewish people in theater, or anywhere else, but that’s missing the point. In “Spamalot,” as in the best comedy of this sort, outrageous humor becomes both a tonic and a salve, even a means of catharsis — and Mr. Rhodes and his team ensure that you’ll laugh your cares away, at least for a couple of hours.