The former world No1 withstood muscle cramps in the second set to outlast his old rival in their 22nd contest on the ATP Tour in Cincinnati.
Murray snatched the opener on a tiebreak and dropped the second amid those physical issues, but would not be denied as he displayed characteristic grit and determination to battle back and come through the deciding set, surviving an earlier break to triumph 7-6 (5) 4-6 6-4.
He will now face compatriot Norrie in the last 32 of the final US Open warm-up tournament, with the British No1 battling to victory over Holger Rune.
Like Murray, Norrie took the opening set on a tiebreak, only for the Danish teenager to come fighing back in the second before ultimately being tamed in a close decider. The 7-6 (5) 4-6 6-4 scoreline was identical to Murray’s.
“The court is pretty lively when the weather is like this. Physically I feel tired,” Murray said after seeing off Wawrinka.
“I had some issues with cramp during the match, the same as Washington, so that is something I need to get on top of.
“I tried to take the ball on a little more and finish points at the net. I wasn’t that successful when I came up, but the intention was there. I was a little more offensive on the second-serve return. Last game I was just fighting, trying to find a way through.”
Murray and Norrie, both named this week in Great Britain’s Davis Cup team for the group stages of the competition in Glasgow next month, have only met once previously on the ATP Tour, with Murray battling to a three-set victory at the China Open in 2019.