Medvedev, ranked sixth in the world, is coming off three straight ATP titles at Rotterdam, Doha and Dubai.
But his hopes that his red-hot form might finally carry him past the round of 16 at the combined WTA and ATP Masters 1000 event in the California desert appeared in serious jeopardy when he twisted his right ankle in a frightening fall in the second set.
Medvedev had been as frustrated as ever with the slow courts of Indian Wells as he fought off four break points in each of his first two service games of the second set.
A changeover rant directed at no one in particular, in which he called the courts "a disgrace to sport," seemed to make Medvedev feel better, and he had given himself a break chance in the sixth game when he turned his ankle as he slid to his forehand side, right leg extended.
His ankle buckled first one way and then the other and he fell to the court in pain.
Zverev, who suffered three torn ankle ligaments at last year's French Open, was quick to rush to his side in concern. After several minutes Medvedev limped to his chair and after having the injury examined and taped by medical staff opted to continue.
"When I rolled my ankle I rolled it pretty hard," Medvedev said. "The moment I rolled it I was like 'OK, I'm going to just stand up and it's going to be fine.'
"And then I was staying on the ground because the pain was only growing and usually that's not a good sign so I thought I'd better stop."
When they did continue, Zverev closed out the game, and they went with serve to another tiebreaker, Medvedev taking a 4-1 lead and closing it with a crisply angled forehand winner to force an unlikely third set.
Medvedev, who appeared largely untroubled by his ankle injury in the third set, broke Zverev for a 2-1 lead.
Crazy match
He twice received treatment on his left thigh - the second time to the consternation of Zverev - but continued to hold serve with ease until he served for the match at 5-4, squandering a match point as he dropped his serve.
Zverev was unable to build on that however, clutching his head in disbelief as he double-faulted on break point in the next game.
Medvedev seized the opening, holding at love to clinch the win after three hours and 17 minutes.
"What was actually surprising was that to move was easier than to walk," Medvedev said. "So that's why when I was walking I was kind of limping but I could actually move kind of fine, probably 95%.
"Definitely a crazy match. I had to adapt my body I was cramping at the end, some strange muscles which usually don't really even work on the court.
"So strange match, but happy to win. I'm going to need some rest."
Medvedev who has now made it to the quarter-finals of every hard court Masters 1000, next faces Spain's 28th-ranked Alejandro Davidovich Fokina, a 6-3, 6-4 winner over Chilean qualifier Cristian Garin.
Garin had toppled world number four Casper Ruud in the third round.
