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8 September 2015, 07:27 | Updated: 30 March 2016, 13:50
Murray was favourite to beat Anderson, having won five of their six previous meetings.
But the big-serving South African played the match of his life to win 7-6 (7/5) 6-3 6-7 (2/7) 7-6 (7/0).
It means Anderson progresses to his first grand slam quarter-final, where he will now play Stan Wawrinka.
Murray led by a mini-break in the first-set tie-break but let it slip and the disappointment seemed to carry over into the second set, as he allowed Anderson to storm 5-1 clear.
The Scot fought back, breaking once, and almost again, but Anderson held his nerve to clinch a two-set lead.
Murray dug deep and sparked hopes of a comeback when he snatched the third set on a tie-break but the world number three left himself too much to do and Anderson closed out in the fourth.
"I felt like I had my opportunities there but didn't manage to capitalise on them - when you're playing against someone as good as him it's tough,'' Murray said.
"In the second set I felt like I was starting to put pressure on him there.
"When I had the break point at 5-3, I had a backhand pass that I really should have made.
"When you're playing against players that are at that level, like him, you need to obviously make them think and then give them a chance to get nervous.''
Nerves could have played their part had Anderson been taken to a decider, particularly given the 29-year-old threw away a two-set lead to lose to Novak Djokovic at Wimbledon in July.
While the 6ft 8in ace-king moves into his first quarter-final, the defeat for Murray brings an end to an impressive run of 18 consecutive appearances in the last eight of major tournaments.
"That's obviously something that is disappointing to lose,'' Murray said.
"Obviously that's many years of work that's gone into building that sort of consistency. To lose that is tough.
"Also, to lose a match like that which was over four hours, it's tough obviously after a couple of tough matches earlier in the tournament. It's a hard one to lose.''
Murray had former Chelsea midfielder Frank Lampard supporting him in his box but the 28-year-old has rarely felt the love in Louis Armstrong Stadium, where he lost to Wawrinka in 2010 and endured struggles against Marin Cilic in 2012 and Robin Haase last year.
The tightness of the court can make it more difficult for players like Murray to defend and the British number one admits the conditions did require an adjustment.
"I practised on the court before the tournament and practised very well on it,'' Murray said.
"Obviously I had some tough losses there, some tough matches, but I've also had some good wins on that court, as well.
"But it's tricky. I've been playing in (Arthur Ashe Stadium) and because of the conditions, Ashe is sheltered from the wind now, a bit slower.
"Armstrong is a tighter court which is very open. You get a lot of wind in there. It's different conditions and something you need to just try and adjust to.''
Anderson had lost his last four meetings with Murray and had failed to win any of his previous seven fourth round matches at grand slams.
"Sitting here having played the match and winning it, it's hard to describe how I'm feeling,'' Anderson said.
"I felt I played one of the best matches of my career.
"To do it at this stage, at this round, obviously to get through to the quarters the first time in a slam definitely means a lot to me.''