Watching the fierce clash over the salad course during a dinner party in “Disgraced,” the rollicking new play by Ayad Akhtar, feels at times like observing a hotly contested game of Twister. As two couples exchange observations about faith and politics in the modern world, the intellectual thickets they find themselves in become increasingly tangled. The language grows more testy, tempers begin to flare, and you have the unsettling sense that someone is going to lose his or her balance and take a hard fall. You’re just not sure who it’s going to be.

The players are a quartet of accomplished New Yorkers of differing races, creeds and, yes, colors, although they have all arrived at the same high plateau of worldly achievement and can agree on the important things, like the tastiness of the fennel and anchovy salad and the banana pudding from Magnolia Bakery. What they cannot agree on — and what will ultimately tear apart at least one of the relationships in the play — is who they really are and what they stand for, once the veneer of civilized achievement has been scraped away to reveal more atavistic urges.

“Disgraced,” which opened on Monday night at Lincoln Center’s Claire Tow Theater in a sleek production directed by Kimberly Senior, is a continuously engaging, vitally engaged play about thorny questions of identity and religion in the contemporary world, with an accent on the incendiary topic of how radical Islam and the terrorism it inspires have affected the public discourse. In dialogue that bristles with wit and intelligence, Mr. Akhtar, a novelist and screenwriter, puts contemporary attitudes toward religion under a microscope, revealing how tenuous self-image can be for people born into one way of being who have embraced another.

The principal characters are a married couple living the Manhattan high life: Amir Kapoor (the spiky, excellent Aasif Mandvi of “The Daily Show With Jon Stewart”), a corporate lawyer at a major firm specializing in mergers and acquisitions, and Emily (Heidi Armbruster), an artist whose profile is on the rise. They share an Upper East Side apartment lusciously upholstered in furnishings from the latest high-end design catalogs. (The set, by Lauren Helpern, offers its own witty commentary on the characters’ identities.)