Fleming and Englishman Jonny Marray had the toughest assignment, taking on the all-conquering, Bryan brothers on the Arthur Ashe Court on Monday evening, with the American twins aiming to become the first pairing in the open era to win all four grand slam titles in a calendar year.
It was a real battle, with two set points going begging in the first -including a routine volley from Fleming which he would normally put away in his sleep - ultimately leading to a 7-6, 6-4 defeat.
Fleming said: "I don't think anyone could have argued with us winning the first set. I missed a really easy volley to break. It's going to be tough to sleep without thinking about that volley.
"We can be proud of the way we played but when it came down to it they just produced the goods on the points that mattered. That's maybe experience of being in these situations more often."
For Murray and Australian partner John Peers, the pain of defeat was more acute. The Swiss/Brazilian pairing of Alexander Peya and Bruno Soares managed to take the first set 6-4 but in the second Murray and Peers forced a tie-break which they won 7-4, mainly thanks to two double faults gifting them the set.
But Murray and Peers failed to create enough chances in the deciding set and went down 6-4.
"It's disappointing, we certainly didn't really return well enough," said Murray. "That meant we weren't able to get in a groove and put us in a position where we could really put some pressure on. Still, I think we can take a lot from this week and if someone had said before the tournament that I would be in the quarter-finals I would have taken it."
Murray has teamed up with Peers for the past seven months after chopping and changing partners and hopes to continue the pairing into 2014.
"John has the potential to be a top-quality player and I hope we can continue together," added Murray. "We have managed to string some decent results together - here we have beaten some good teams - so I am confident looking forward. We are a team on the up."
The dream is also over for Lleyton Hewitt after a dramatic four-hour, five-setter against Mikhail Youzhny ended with the Russian eventually winning 6-3, 3-6, 6-7, 6-4, 7-5 in one of the matches of the tournament.
Australian Hewitt was in a great position to reach his first grand slam quarter-final since 2009, having fired himself into a 5-2 lead in the final set before Youzhny superbly countered to win five games on the spin to reach the last eight in stunning fashion.
Novak Djokovic booked his place in the quarter-finals by destroying the hapless Marcel Granollers 6-3, 6-0,6-0 in an embarrassingly one-sided match.
Meanwhile, world No.2 Victoria Azarenka recovered superbly to see off Ana Ivanovic and reach the quarter-finals with a 4-6, 6-3, 6-4 victory to set up a last eight match with Daniela Hantuchova.
The Belarusian remains the biggest threat to usurping champion Serena Williams and said:" I think Daniela is playing incredible tennis right now, you know, really going so deep in the tournament.
"She's a player who has been in these situations before with a lot of experience, a lot of motivation right now. And she has the type of game that she's a shot-maker, so it's going to be another tough battle of really taking your chances, you know, playing your game."