Murray's meeting with Wawrinka has been picked as the second match of the night session on the main Arthur Ashe Stadium in New York, following a women's match featuring Serena Williams.
Murray had to dig deep to secure a 6-7 4-6 7-6 6-1 6-3 win against Melzer, and the 21-year-old believes the victory was the finest of his career - even beating his five-set Wimbledon win over Richard Gasquet this year.
"I was in danger in both matches, but I felt like Melzer was playing so well that it was going to be really tough for me to get back into the match," he said.
"Gasquet was obviously playing great too, but I still felt like I had a shot of getting back in there. And I had known in the past that he had struggled to close big matches out and in five-set matches, so that made it a little bit easier than the match against Melzer.
"I'm not that good of a player that I can just blitz guys. But I said it was going to be a difficult match, Melzer's very talented, so I was delighted to come through it."
Murray will now look to reach his second Grand Slam quarter-final of the year with victory over Wawrinka, who also had to come through a five-set epic in the last round, beating Flavio Cipolla 5-7 6-7 6-4 6-0 6-4.
The pair have played each other six times in the past and have three victories apiece, with Murray winning the last encounter 6-2 0-6 6-4 at the Canadian Masters.
In fact, the Scot has a 3-1 winning record on hard courts, including victory in the final of the Qatar Open this year.
And Murray believes he can continue his march towards a first grand slam career win with victory over the Swiss, saying: "I'm going to give it my best shot and try to win.
"I might not necessarily win the tournament, but as I've gone into it with the intention of doing so it might not come as a surprise if I go deep."