Zverev struggled with his fellow German's chip and charge game in the first grasscourt tournament of the season ahead of Wimbledon.
A French open quarter-finalist, Zverev, saved four set points on his serve at 5-3 in the first but could do nothing to stop the 34-year-old Brown from serving out the first set a game later.
The world number five, who has never won a title on grass, improved in the second and quickly broke in the second game to race 5-2 clear.
Yet Brown continued to mix it up and broke back to level at 6-6 before his opponent, who served 14 double faults, clinched the second set 7-3 in the tiebreak.
Brown saved three break points at 3-3 in the third and never looked back, racing to finish it on his first match point.
"It's a big win. The last few years have been a bit up and down with injuries. So I am very happy and especially that it happened in three sets which was mentally very challenging," said Brown.
"I stayed cool, tried to play deep and am happy to have won it."
He will now play Canadian teenager Felix Auger-Aliassime in the last eight after the seventh seed beat France's Gilles Simon in straight sets.
Italian Matteo Berrettini earlier caused the first major upset of the tournament, beating second seed and French Open quarter-finalist Russian Karen Kachanov 6-4 6-2 in just over an hour.
He will next face American Denis Kudla, who earned the biggest win of his career with a 7-5 6-7(3) 7-6(3) victory over world number 16 Gael Monfils.
(Reporting by Karolos Grohmann; Editing by Toby Davis)
