Despite the loss, Mischa said it was a match he would always remember.
"After the coin toss I had tears in my eyes because it felt so special," he added.
"I wondered what are my parents thinking right now. I needed a few seconds to bite my tongue and focus."
The younger Zverev said he hoped to see his brother across the net again in the future.
"I hope this is not the last time. I hope we play a final one day," said Alexander, the tournament's defending champion.
Scot Andy Murray broke down in tears after coming from behind to defeat Marius Copil 6-7(5) 6-3 7-6(4) in a marathon third-round match that lasted more than three hours.
Murray, the former world number one who is attempting to return to the top of the tennis world after undergoing hip surgery in January, overcame 20 aces by Copil to advance to the quarter-finals, where he will face Australian Alex de Minaur.
Earlier in the evening de Minaur benefited from a poor serving performance by Chung Hyeon, who double faulted seven times and won just 56 percent of his first serves, to advance 6-2 4-6 6-2.
Second seed John Isner was upset by fellow American Noah Rubin 6-4 7-6(6) in the second round.
Isner, last month's Atlanta Open champion, appeared fatigued during the match, perhaps owing to a busy July where he played 10 matches consisting of 34 sets and 403 total games.
The win over world number nine Isner was the 22-year-old Rubin's first over a top 10 opponent.
Due to rain delays earlier in the tournament, however, Rubin had to play 20-year-old Russian Andrey Rublev later on Thursday where he fell 6-3 6-2.
Stefanos Tsitsipas of Greece, the 10th seed, capitalised on a strong service game that included eight aces to earn a 6-3 6-4 win over Australian James Duckworth.
Tsitsipas will face third seed David Goffin in the quarter-finals after the Belgian beat American Frances Tiafoe 6-0 6-3.
(Reporting by Rory Carroll; Editing by Peter Rutherford and Sudipto Ganguly)
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