The British number one, who has replaced Roger Federer as the second best player in the ATP World Tour rankings following his Rogers Cup triumph, will next face Grigor Dimitrov after triumphing 6-4 7-6 (7/1) against the soon-to-retire Fish.
The 33-year-old from Minnesota recently spent 18 months out of the game due to a heart condition and will step away from tennis for good after the US Open, but held his own in the first set and managed to force the second-set tie-break having surrendered a 4-3 lead.
Murray paid tribute to Fish - a former world number seven - in the aftermath of their late-night encounter in Ohio.
Posting a picture of his opponent, he tweeted: “With @MardyFish after our match tonight/this morning! One of my trickiest opponents good Person, fun guy
“Who’s been very unfortunate these last few years... All the best for the us open and your life after tennis!”
Murray’s next opponent Dimitrov may have enjoyed the benefit of a day’s rest having beaten Vasek Pospisil in the second round on Tuesday, but he has only got the better of the counter-punching Scot twice, knocking him out of Wimbledon in 2014 and winning a semi-final in Acapulco earlier in the same year.