Carreno Busta broke Zverev to open the second set and built a 3-1 lead before the German rallied, setting up a quarterfinal match against Jannik Sinner.
"I have lost focus a little bit in the second set," Zverev said.
He then played more aggressively, adding: "I am not going to win big titles just by pushing the ball."
"I really had to work hard for this win," Tsitsipas said.
Sinner beat eighth-ranked Andrey Rublev 5-7, 6-1, 6-3 to advance, winning six straight games in the second set and dominating the decider before wrapping it up with his seventh break.
"I tried to win every play, every point," said the 20-year-old Italian, who reached the quarterfinals in Miami last month before retiring from the match against Francisco Cerundolo.