Team World had overturned a 3-1 overnight deficit in the first two matches of the day at London's O2 Arena as all three singles matches went to a deciding tie-break.
Learner Tien beat Flavio Cobolli before French and US Open champion Alexander Zverev lost out to Alex De Minaur.
Alcaraz was beaten by Fritz in last year's Laver Cup as Team World stormed to victory in San Francisco.
The Spaniard exacted his revenge but was pushed all the way before edging a classic 6-3, 4-6, 13-11.
"Two great and tough losses today and obviously I was thinking a little bit about last year, (when) we lost every match on Saturday," said Alcaraz.
"Coming into the match, I was trying not to let it happen again. So I was kind of nervous at the beginning, I'm not going to lie. But really happy that I took the win at the end."
Zverev teamed up with Casper Ruud to give Europe the advantage heading into Sunday with a 6-4, 7-6 (7) doubles victory over Alexander Bublik and Brandon Nakashima in the final match of the day.
Each match on Sunday is worth three points with the first side to 13 the winners.