Murray and Peers were both through to their first Grand Slam quarter-final in men's doubles but were underdogs going into the clash.
Austrian Peya and his Brazilian partner Soares edged the first set but Murray and Peers fought well to take the second to a tie-break and won it thanks to an untimely double fault from their opponents.
The decider was a tight affair but it was the British-Australian pair who were having to withstand most of the pressure and that told with Peers serving at 4-5.
The Australian saved two match points but not a third, and it is Peya and Soares who go forward to the semi-finals following the 6-4, 6-7 (4/7), 6-4 win.
On Monday night, Fleming and Marray lost a close tussle with Bob and Mike Bryan, leaving Dom Inglot, who plays with Filipino Treat Huey, as the only British player left in doubles competition.
Inglot and Huey face 10th seeds Ivan Dodig and Marcelo Melo in the last eight today.
While his brother dropped out of the US Open, Andy Murray has put himself forward to play on all three days in Great Britain's Davis Cup World Group play-off against Croatia next week.
The Scot is set to play his first Davis Cup tie in two years after being named in the British team alongside Dan Evans, James Ward and Fleming.
But there is no place for Fleming's doubles partner, and last year's Wimbledon champion, Jonny Marray.
That implies that Murray is keen to reprise his doubles partnership with childhood friend Fleming, with whom he reached the Masters final in Montreal last month.
Marray's omission is a second disappointment in a matter of hours for the Yorkshireman.
Captain Leon Smith has, however, left the option open to bring in another doubles player after the US Open has finished.
Smith said: "We are delighted to be sending our strongest team to Umag in what will be a very tough tie.
"Croatia are former winners of the Davis Cup and currently ranked No.11 in the world but, as we proved in Coventry back in April, we can upset the odds.
"The recent success of our players has inspired more people into our sport and this is another great opportunity."
Should Murray reach the final of the US Open, he will have three days to travel and adjust to clay before potentially playing three five-setter in as many days.
Croatia will be led by world No.38 Ivan Dodig but their No.2, 187th-ranked Antonio Veic, has not won a tour match all year.
In singles action last night at Flushing Meadows, Lleyton Hewitt's excellent run ended in heartbreak as he lost in five sets to Russia's Mikhail Youzhny in the fourth round, while Novak Djokovic thrashed Marcel Granollers of Spain 6-3, 6-0, 6-0 to progress.
Victoria Azarenka battled through to beat Ana Ivanovic in the fourth round over three sets.