Music is the universal language of mankind, as Longfellow famously noted, and nowhere is this more evident than in live performance. Watching great musicians who enjoy a genuine rapport can be like observing close friends, or lovers, sharing silent intimacies; though this is particularly true in jazz, where improvisation is key, it can manifest itself in all genres.

Such chemistry is front and center in the new musical “Buena Vista Social Club,” inspired by the stories and featuring the music of that titular group, a collective of mostly Cuban, seasoned musicians that rose to acclaim in the late 1990s. Named after a club that had thrived in Havana about a half century earlier, the ensemble played standards of their culture; their work was captured on an eponymous, Grammy Award-winning album and an Oscar-nominated documentary film directed by Wim Wenders. 

Mr. Wenders used interviews and performance footage to pay homage to the musicians, several of whom had been coaxed out of retirement or near-obscurity with the support of American guitarist Ry Cooder, who produced the album. In the stage musical, librettist Marco Ramirez introduces us to characters based on the artists, whom we alternately observe in 1956, before Fidel Castro’s revolution and reign dampened their career prospects, and in 1996, as their country suffers in the wake of the Soviet Union’s collapse.

We meet guitar player Compay Segundo, pianist Rubén González, and singers Ibrahim Ferrer and Omara Portuondo — all named after prominent participants in the collective — both as rising young artists and as they grapple with advancing age and deteriorating conditions around them. Luis Vega plays Juan De Marcos, apparently inspired by Juan De Marcos González, who helped Mr. Cooder put the group together — presented here as an eager young producer, determined to capture the elders’ long-neglected gifts. 

Mr. Ramirez’s book, like many of those crafted for jukebox affairs, is rather short on imagination and nuance. The young Omara, for instance, is pursued by one record company who wants to play down her heritage and another that shuns Ibrahim because of his darker skin color, and both dilemmas, while substantial, are addressed in an obvious and occasionally pedantic manner.

All is forgiven, however, once the music kicks in, which it does in joyful abundance. A virtuosic band is prominent on stage through much of the performance; a showcase early in the second act features an overlong guitar solo, where the player’s transparent enjoyment of his own prowess, egged on by the others, may remind you of certain MTV-era rock bands — even if the playing is more graceful.

Otherwise, the musical conversations unfolding in “Buena Vista Social Club” are glowing testaments to Longfellow’s oft-quoted words. You needn’t understand a word of Spanish to enjoy the impassioned synergy and playful give and take during enchanting performances of boleros and songs from other Cuban genres, such as son and guajira. 

This sense of camaraderie extends to the vocalists; standouts include Kenya Browne and Natalie Venetia Belcon, respectively cast as the young and older Omara, both of whom sing with a warmth and a distinctive textural richness that fit the material like a velvet glove. The Cuban recording artist Mel Semé, playing the elder Ibrahim, brings a similar affinity, along with a gentle gravitas that makes the character’s difficult journey particularly affecting.

Co-choreographers Justin Peck and Patricia Delgado, who worked together on Steven Spielberg’s screen adaptation of “West Side Story” — Mr. Peck also won a Tony Award for the most recent Broadway revival of “Carousel” — once again inject their ballet expertise into intensely theatrical representations of Latin dance that mirror and enhance the moods expressed by the musicians and actors. Exuberant production numbers are offset by dancers stretching and swooning emotively, sometimes in groups or pairs, behind the principals.

Arnulfo Maldonado’s spare but inventive set design facilitates the space adjustments required to accommodate such activity alongside a sizable band on the theater’s relatively small stage. Whatever its flaws, this “Buena Vista Social Club” is bound to fill your heart accordingly.