Reigning French Open and US Open champion Zverev lost to De Minaur 2-6, 7-6(5), 11-9, after American Tien beat Cobolli 6-3, 2-6, 10-7, allowing Team World to wipe out Europe's 3-1 overnight lead.
With two points on offer for each match win on the second day in a race to 13 points for the championship, De Minaur made the most of the opportunity despite a first-set loss in which Zverev dominated with strong forehands near the sidelines.
Zverev was 5-4 up and serving for the second set when he committed two double faults that allowed the Australian to bounce back with a break and win the set on a tiebreak.
"Check his oil," World captain Andre Agassi told De Minaur, asking him to force the 29-year-old Zverev into tiring, long rallies.
The advice paid off in the 10-point match tiebreak, with De Minaur using repeated cross-court shots to thwart Zverev when the German was 9-8 up, before getting match point and securing victory with a precise backhand down the sideline that Zverev could only hit back to the net.
"It was a tough battle, I left it all out there," said De Minaur after his second straight Laver Cup win over Zverev.
Earlier in the day, Agassi's American compatriot Tien beat French Open finalist Cobolli in another nervous tiebreak.
Tien controlled the contest from the net as he eased to a 4-0 lead in the first set. Cobolli fought back with a break and, although it did not save the first set, the Italian carried the momentum into the second, getting two breaks in a row on his way to winning it.
But the 20-year-old Tien, playing in his first Laver Cup, targeted the baseline to frustrate Cobolli in the match tiebreak and won a 34-shot rally to clinch victory.
"I think in this format, no matter what the score is... the match can turn on a dime," Tien said.
Europe will hope to fight back when world number three Carlos Alcaraz plays Taylor Fritz later on Saturday, followed by a doubles match where Zverev and Casper Ruud will team up against Team World's Brandon Nakashima and Alexander Bublik.