It was as muggy as Miami inside a half-full Jazz Café, and Tiempo Libre were in their element. “Que rico!” yelled pinstriped frontman Xavier Mili, rolling his Rs and thrusting his hips, as musicians on everything from horns, congas and kit drums to flute, keyboards and bass guitar played upbeat Cuban music with a sometimes cheesy, often soulful American twist.

All bespoke suits, diamanté ear studs and good-natured showmanship, the Cuban group are big news in the US, where they were founded in 2001 after the seven members graduated from the Cuban conservatory system and one by one, found their way to Florida’s capital. Six albums and three Grammy nods later they probably are the most authentic all-Cuban timba band in the US.

Britain, where aficionados of salsa cubana prefer their music direct from Havana, was always going to take some convincing. But Tiempo Libre’s take on timba — that fierce mix of pop, R&B, Latin jazz and Cuban son — proved all the more rewarding for its sophisticated ideas and tight-knit arrangements.

Having established their jazz chops with an instrumental steered by musical director Jorge Gómez, a man given to playing his Yamaha keyboards with one knee on the ground, they went classical. The opening lines of Bach’s Minuet in G folded into a guaguancó, with dreamy jazz trumpet from Raul Rodriguez; Sonata in D Minor became a cha-cha-cha.

Then came the cheese: a singalong Guantanamera. A blonde salsera, serenaded. But by the time the bolero-son Lagrimas Negras changed gears into a chewy timba workout buoyed by conguero Leandro Gonzalez, everyone was moving; a fast-paced Candela kept things hot. An intimate UK debut, then, by a world-class act. They play Latitude Festival this weekend.