One break in each set was enough for the German to clinch his first win in three career attempts against the Croat.
Zverev advances to a Friday semi-final against Spanish 16th seed Carreno Busta, who survived a match point to upset South African sixth seed Anderson 6-4 5-7 7-6(6).
The other semi-final pits Argentine fifth seed Juan Martin del Potro against American 14th seed John Isner.
Zverev did not use the power failure as an excuse for his slow start.
"I didn't particularly care," he said, adding that he spent the time watching an NBA game on a locker room television.
"I think I lost the first six points of the match. It wasn’t the best start. I’m just happy to get through."
For Carreno Busta, who arrived on court in fine form having dropped only 12 games over his previous tee matches, the win was his first in five meetings against Anderson and avenged his fourth-round Indian Wells loss to the world number eight.
"It was an amazing match, no?" Carreno Busta said.
"He played some unbelievable points, passing shots.
"I lost to him, 7-6 in the third set (at Indian Wells). And today, well, at the beginning of the match I started playing really good, returning good, serving good, being very aggressive."
Anderson saved a pair of match points in the second set and shifted momentum in his favour but Carreno Busta, seeded 16th at Crandon Park, regrouped in the decider where the two players stayed on serve until the tiebreak.
Carreno Busta was staring at defeat when 6-5 down in the tiebreak but was let off the hook when Anderson missed with a forehand.
From there the South African pushed a backhand volley wide and then mishit a second serve that handed the win to Carreno Busta after two hours, 42 minutes.
Anderson, whose year already includes an ATP Tour title at the New York Open and runner-up finishes in Acapulco and Pune, has now lost all 10 of his quarter-final matches at the Masters 1000 level - the highest-tiered tournaments after the four grand slams.
(Reporting by Andrew Both in Cary, North Carolina and Frank Pingue in Toronto; Editing by Peter Rutherford)
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