Italian Cecchinato won in four sets, scrambling over the line in an epic tiebreak on his fourth match point.
"I think that's going to be the number one priority for me now to kind of step out from the tennis for a little bit, try to watch some matches and try to progress and get better".
The 23-year-old American has now reached at least the last-eight of all four majors, after ending the dream run of Buzarnescu, who had never won a Grand Slam match before the tournament.
"It was an fantastic, high-level match", said Zverev.
German second seed Alexander Zverev and Novak Djokovic edged closer to a possible French Open semi-final clash of generations, with the youngster edging out Karen Khachanov and the former champion seeing off Fernando Verdasco in the last 16 on Sunday.
World No. 29 Dzumhur had served for the match in the fourth set, but Zverev clung on and battled to a 6-2, 3-6, 4-6, 7-6 (7-3), 7-5 victory. "So it's going to be an unbelievable challenge for me".
US Open champion Sloane Stephens cruised past Estonian Anett Kontaveit 6-0, 6-2 to book a last-eight spot for the first time.
His climb up the rankings and his efforts here are all the more remarkable given what happened to him in 2016, when he was embroiled in a match-fixing scandal in Italy.
Mauricio Pochettino opens door to move to Real Madrid
This is Real Madrid we're talking about though so of course, it wouldn't really be a surprise to see anyone turn up in that hotseat.
But he has never reached a Grand Slam quarter-final, although the longer best-of-five format has worked twice in his favour now at Roland Garros.
Like her compatriot and good friend Keys, US Open champion Stephens is in the quarter-finals for the first time and hoping to set up a repeat of last year's final in NY.
Earlier in the day Keys, the 13th seed, reached the semi-finals for the first time by seeing off Kazakhstan's Yulia Putintseva 7-6 (7/5), 6-4.
Since then, she is 19-6 and she's seeking her second career Grand Slam win. "There was another year of work where I improved and developed my game", said the 24-year-old.
Stephens, the 10th seed, had previously made the fourth round for four successive seasons and will next play either Australian Open champion and second seed Caroline Wozniacki or Russia's Daria Kasatkina.
Djokovic, bidding to reach a 32nd Grand Slam semi-final, hit just three winners in the first set.
"Honestly, the (US) Open feels like it was 12 years ago at this point", said Keys, who is yet to drop a set in the tournament. I know it's hard right after the match, but I need to - I mean, I have always been a positive thinker, and I want to keep that on the same level right now.
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