Devotees of the Broadway musical have been gasping for a saviour. Risk-takers such as the Green Day-scored American Idiot can't survive (it closes at the end of April), and corporate fiascos such as Spider-Man: Turn Off the Dark threaten to turn the Great White Way into a global joke. That's why The Book of Mormon, gleefully subversive and artfully crafted, is being hailed as the second coming; this new work by South Park creators Trey Parker and Matt Stone (and composer-lyricist Robert Lopez from the naughty-puppet hit Avenue Q) is a good old-fashioned song-and-dance spectacle that happens to include wildly offensive jokes about Aids in Africa and the theological kitsch that is Mormonism.

If you're surprised to hear that Parker and Stone are responsible for re-energising Broadway's hopes, you haven't been following their career. The team have been honing their razzle-dazzle chops over two decades. Their first major effort, Cannibal! The Musical, was filmed in 1993, and, in 1999, South Park: Bigger, Longer & Uncut was aptly (if cheekily) praised as the year's best new musical. More recently, Team America: World Police paid snarky homage to Rent with the parody ballad "Everybody Has Aids". These showtune-humming pranksters were destined to mock the Church of Jesus Christ of Latter-day Saints in song – an institution that, like the Broadway musical, is a singularly American invention.

Starting off in the Mormon mecca, Salt Lake City, Utah, the story follows a mismatched pair of proselytisers, Elders Price (Andrew Rannells) and Cunningham (Josh Gad). The former is the clean-cut ideal of an LDS doorbell-pusher: white-bread, well-groomed and safely asexual. Cunningham, however, is a fat, dim-witted man-child who confuses Mormon mythology with The Lord of the Rings.

Despite Price's hope for missionary work in Orlando, Florida, the two are ordered to save souls in war-torn, poverty-stricken Uganda. Their evolving friendship lays the emotional foundation for the show, and gives even the cruellest jokes about racism and homophobic self-loathing a sweet, innocent finish. That human dimension reminds you that the comic genius of South Park (heading into its 15th season) relies on children blinded by naivety, but who see through society's lies.

Likewise, by smashing together cultural extremes – prim, uber-Caucasoid Mormons and long-suffering, hope-starved Africans – the creators lampoon western illusions about that complex continent (the anthem "I Am Africa" is sung by distinctly pale cast members), while scoring laughs off the sort of horrors that should never be put on a Broadway stage ("I have maggots in my scrotum" is a recurring lament by one villager). We chortle disgustedly at an African man who thinks raping a baby will cure his Aids (a documented crime), but truly grotesque is the notion that a couple of Bible-toting white boys can be of any real help.

Religion, the creators firmly point out, is showbiz, and they systematically dismantle the absurdities of Joseph Smith's 19th-century cod revelation through the intoxicating frivolity of musical conventions. Of the dozen or so classics referenced in the pastiche score, or by sight gag and laugh line, you can count The Sound of Music, Wicked, Jesus Christ Superstar, The Music Man and (naturally) The Lion King. Now The Book of Mormon – aggressively hilarious, blasphemous and almost indecently entertaining – has grabbed a spot in that canon. For those of us who love a well-made musical with satirical bite, the show is manna from heaven.

Until 19 November

David Cote is chief theatre critic of Time Out New York

• This article was amended on 10 June 2011. The original said the Church of Latter-Day Saints. This has been corrected.