The match was interrupted for 90 minutes in the seventh game of the second set by rain and had two more five-minute pauses as drops were dried from the court.
French Open champion Zverev had a chance to close out a victory at 6-5 in the second-set tiebreaker but stuffed a volley into the net.
The German again gained an edge in the deciding set with a break for 4-2, but Paul broke back in the next game and again for a 5-4 lead.
He served out the victory with confidence for his third win over Zverev in their last four meetings.
“I’m not sure how I did it,” Paul said. “I put some returns in, he gave me some looks at second serves and he threw in a few doubles (faults). That always helps a ton.
“I went after the ball a bit more, I was having fun out there. I hope I can keep building on this and peak at the US Open.”
Paul next faces Roland Garros runner-up Flavio Cobolli, who laboured two and a half hours to engineer a turnaround against red-hot Spaniard Rafael Jodar 4-6, 7-6 (7/3), 6-3.
Cobolli has won all three of his matches in Cincinnati in three sets.
The durable Italian reached his second Masters-level quarter-final of the season after Madrid, and he prevented 19-year-old Jodar from reaching an eighth quarter-final in nine events.
“At the end of the second set and start of the third I told myself to believe, try on every point,” Cobolli said. “I knew I had a chance.
“Rafa has had an unbelievable season but maybe he was a bit tired. At the end I deserved to win as I was fresher than him.
“I pushed to my limit to get through. Sometimes I have to remind myself that I’m really strong.”
Thiago Tirante, who upset Novak Djokovic in the second round, continued his surprise run at the last big event prior to the August 30 start of the US Open.
He recovered to defeat Jakub Mensik 5-7, 6-4, 6-4 and book his first Masters quarter-final.
Australian Alex de Minaur was no match for Frenchman Arthur Fils, who cruised to a 6-3, 6-4 victory over the fifth seed. (AFP)
