Soon after the beginning of “All In: Comedy About Love by Simon Rich,” the comedian and actor John Mulaney reads a letter by Mr. Rich, with whom he worked years ago as a writer on “Saturday Night Live,” explaining the mission of this 90-minute piece. “Deep down it’s meant to tell one simple story,” Mr. Mulaney relays, “about how the most important part of life is who we share it with.”

If that description suggests a sentimental journey, think again. Mr. Rich — who in recent years has served as executive producer on hit TV series such as “Veep” and “Succession” and is the younger son of a former New York Times theater critic and columnist, Frank Rich — has collected numerous credits as an author and screenwriter himself, among them short stories published in the New Yorker.

“All In,” which will feature a rotating series of stage and screen stars during its limited run — Mr. Mulaney is currently joined by Fred Armisen, Renée Elise Goldsberry, and Richard Kind — consists of a series of vignettes adapted from Mr. Rich’s stories, and their general sensibility exists at the intersection of “SNL” and the New Yorker. Varying in length from short and sweet to overextended, they’re literate, irreverent, and goofy in that knowing way one associates with the kind of bright, impeccably educated young men (and sometimes women) who find success writing for late-night television shows and prestigious magazines. 

Some of the best material arrives in the show’s first 20 minutes. In the opening segment, called “Guy Walks Into a Bar” — titles are provided above David Korins’s set, which suggests a hipster’s apartment, with plush chairs posed in front of a stuffed bookcase — Mr. Mulaney delivers, with perfect timing, an account of a genie with poor hearing; one character is a bartender who’s rather insecure in his masculinity, while another is a 12-inch pianist.

“Dog Missed Connections” continues in this blue vein, detailing personal ads placed by canine contenders. In a relatively tame entry, one four-legged suitor appeals to a bitch who got away: “I managed to mount you. … You shook me off, though, and ran away. I’m interested in getting to know you a little better. I’ll be at the dog run tomorrow morning. Hope to see you there.”

Smart but protracted tales involving pirates and baby detectives follow, along with “Case Study,” which concerns a stuffy British doctor, expertly played by Mr. Kind, who makes the mistake of introducing his wife to Joseph Merrick — better known as the Elephant Man, presented here as something of a lady-killer, and portrayed by a hilariously low-key Mr. Armisen. “New Client” casts Mr. Kind as an elderly talent scout who cheats Mr. Armisen’s Death by offering him a career in show business. 

In between skits, the Bengsons, a married duo of composers and performers with extensive experience in theater, offer playful, wistful songs that alt-pop hero Stephen Merritt recorded as the Magnetic Fields, described in Mr. Rich’s letter as “my favorite band … who inspired me to start writing love stories when I was a teenager, instead of dating anybody.”

The actors, too, all fare well under Alex Timbers’s characteristically nimble direction. (Performers scheduled to appear later in the run include Lin-Manuel Miranda, Jimmy Fallon, Tim Meadows, Andrew Rannells, and Annaleigh Ashford.) For me, it was especially delightful to see Ms. Goldsberry, a “Hamilton” alumna whose exquisite singing and acting (not to mention her ageless beauty) should have earned her more leading roles on Broadway, showcase her considerable comedic prowess, in roles that include a little girl who falls in with pirates and a budding gumshoe who’s even younger.

Ms. Goldsberry gets to close “All In” with “History Report,” a monologue that features Mr. Rich’s most transparently personal writing here. Both earnest and naughty — and evocative, in its tone and subject matter, of something you would hear on NPR, another good reference point for the show — it’s a fitting topper for a clever confection that, in the end, doesn’t overstay its welcome.