The 2012 champion earned his first win over a top-10 player since his Wimbledon triumph last year to set up a clash with world number one Novak Djokovic.
Murray said he hoped Wednesday's quarter-final between the two would go the same way as both their battle for the Wimbledon title and the 2012 U.S. Open final, which he won.
"It'll be a tough match," the Scot said after a two hour, 35-minute battle in humid conditions with temperatures close over 32 degrees Celcius (90 degrees Fahrenheit).
"We've had a lot of long ones, normally a lot of rallies, long points. We played a lot of long ones here a couple of years ago and I have great memories of that match.
"It's going to be very hard but if I play well, I'll have a chance."
Eighth seed Murray dominated the first set, taking it thanks to one break in the 12th game.
But the 27-year-old had to come from 4-2 down in the second set to go 2-0 up, smashing a forehand down the line off the Tsonga serve.
Again Murray found himself behind in the third and if ninth seed Tsonga had converted one of three break points at 2-0 he would have faced an uphill task to get back into the set with a double-break against him.
But the Scot raised his game when he needed to and secured his place in the last eight.
"I feel like I'm playing well," he said. "I'm going to have to play extremely well to win the event or even win the next match.
"A week ago I didn't feel great at all and a week later I'm in the quarters so a lot can happen."
(Editing by Gene Cherry/Nick Mulvenney)
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