World No. 3 Alcaraz, competing in his second tournament after the US Open following a four-month injury lay-off, joined Mensik to beat Alexander Bublik and Taylor Fritz 6-4 6-4.
The Spaniard and Czech, who also partnered at last year's Laver Cup in San Francisco, fended off three early break points before Alcaraz found his range and fired an overhead smash to secure the first break.
They broke again early in the second set and held firm from there, with Mensik serving out the match, with Alcaraz sealing victory at the net.
"I'm just trying to have fun in every place that I go and on every court that I play on," Alcaraz said.
"I don't usually play doubles ... I just try to enjoy and be better at things. I use doubles to try and improve my tennis, a couple of adjustments. Having a partner like Jakub, it's really easy to have fun on the court."
The victory gave Europe three points from Friday's four matches, after Norwegian Casper Ruud got them off to a winning start by edging Francisco Cerundolo 6-7 (7-4) 6-4 10-8.
Brandon Nakashima levelled the contest for Team World by beating Mensik 6-1 3-6 10-7, before Rafael Jodar restored Europe's lead with a commanding 6-2 6-3 victory over Bublik in the Spaniard's Laver Cup debut.
Jodar, who was watched from the stands by Roger Federer and John McEnroe, said he was thrilled to have the tennis greats at his first appearance in the competition.
"It's a pleasure that they are in the stands watching me," Jodar said.
"It's a pleasure to see them watching these types of matches."
The three-day Laver Cup pits six players from Europe against six from the rest of the world, with victories worth one point on Friday, two on Saturday and three on Sunday.
Fresh from inspiring Australia at the Davis Cup, Alex de Minaur is part of Andre Agassi's Team World.
De Minaur, ranked world No.9, has a tough singles battle on Saturday (local time) against newly-crowned US Open champion Alexander Zverev, who is representing Team Europe.
They are the second match on court following Flavio Cobolli (Team Europe) and Learner Tien (Team World).
In the night session Alcaraz meets Fritz the doubles clash with Europe's Zverev and Ruud up against the World's Nakashima and Bublik.
