The Australian duo are now through to the semi-finals of the mixed doubles, and within sight of a medal.
After Barty and Peers took the first set 6-4, Sakkari and Tsitsipas hit back to claim the second by the same margin.
The Aussies then closed out the match tie-breaker 10-6, to setup a meeting with the Russian pair of Anastasia Pavlyuchenkova and Andrey Rublev in the semi-finals.
Meanwhile, Novak Djokovic has destroyed Kei Nishikori in the quarter-finals of the men's singles.
The top-ranked Serb never gave Nishikori a chance in a 6-2, 6-0 rout of the home favourite from Japan to reach the semi-finals.
"He was defending amazing today — everything deep — and I was trying to stay with him but I couldn't," Nishikori said. "I thought I was playing OK but my serving was bad today and he was attacking every (time)."
The dominant performance advanced Djokovic into the medal rounds and — more importantly — moved him one step closer to finishing off this segment of his Golden Slam bid.
Djokovic will next face either Alexander Zverev of Germany or Jeremy Chardy of France. The other semifinal match will feature Karen Khachanov of ROC against Pablo Carreno Busta of Spain.
Khachanov beat Ugo Humbert of France 7-6 (4), 4-6, 6-3 and Carreno Busta defeated second-seeded Daniil Medvedev of ROC 6-2, 7-6 (5).
"The way he's played today, this week and last couple months, it's been amazing," Nishikori said.
As it's been for months, it seems that Djokovic, who has not dropped a set in the tournament, is getting better match by match — so much so that it looks like he's hardly being tested.
"Matches are not getting easier but my level of tennis is getting better and better and and I've done that so many times in my career," Djokovic said. "I'm (the) kind of player that the further the tournament goes the better I'm feeling on the court and that's the case here."
Djokovic added that it was his "best performance" of the tournament.
"Against (a) very good opponent," he said. "I felt like I had an answer for everything he had.
Play started four hours later than on previous days after organisers finally acted upon a request from players to avoid the hottest part of the day.
"Playing after after 5:00 is completely different," Djokovic said. "Still very, very humid, you sweat a lot, but you don't have the heat, you don't have the sun in combination with the humidity."
- With AP
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