They had started Sunday 8-4 down, with three points on offer for each win in the final day’s matches in the bid to hit 13.
The action resumed with Alcaraz and Casper Ruud reducing the deficit via a 6-2 7-6 (6) doubles victory over Ben Shelton and Frances Tiafoe, before Shelton defeated Daniil Medvedev 6-7 (6) 7-5 10-7 to put the rest of the world 11-7 up.
Team World were then on course to take the trophy for a third successive edition with Tiafoe a set and a break up against Alexander Zverev, only for the German to fight back to win 6-7 (5) 7-5 10-5 and take things to a decider.
Alcaraz subsequently overcame Fritz 6-2 7-5 to spark celebrations among the European team.
And the Spaniard – who on Saturday had hit out at the packed tennis calendar, saying of the sport’s governing bodies that “probably they are going to kill us in some way” because of the physical demands exerted on players – said in his on-court interview: “Obviously we came here with a goal, all of us, to win the Laver Cup.
“I think we are really happy about reaching our goal. I think all of us did pretty good work over the days.
“It’s been a really tough one, we almost lost. But Sascha (Zverev) came with really good tennis and gave me the chance to win the cup. So we are really happy with everything.”