The 19th seed did not have an easy tournament start against the world number 153, whose own first serve percentage matched that of Zverev, and with the German getting just two break opportunities in total in the first two sets.
He next plays another lower-ranked player in Japan's world number 116 Yosuke Watanuki with little time to rest as the schedule has been reworked following all the rain delays.
"He was a very tough opponent and the rain was a very tough opponent," said Zverev, whose match was one of the last first- round clashes to finish. "I'm just happy to be back at Wimbledon."
He missed the tournament last year following his ankle injury at the French Open.
"I missed it last year with a bad injury so just to be on this court I'm happy to be here with 10,000 people. I know Wimbledon is the most historic tennis tournament in the world and maybe the most historic sporting event in the world.
"I play again tomorrow and the day after, well if I win of course. It's going to be a tough few days with back-to-back matches and it is never easy physically - I'm just happy to be playing tennis now," Zverev said.
He did snatch a break at 3-3 in the first set before serving it out and both players held serve until left-hander Brouwer carved out two set points at 6-5.
Zverev, a former world number two looking to hit top form again after an ankle injury derailed his 2022 season, saved them both to force a tiebreak, winning it with another booming ace.
Both players comfortably held serve in the third to take it to another tiebreak and Zverev fired down his 20th ace to seal victory on his third match point.