Murray fights back to reach Dubai second round
The former world number one, who snapped a six-match losing streak in Doha last week before going out against Czech teenager Jakub Mensik, won 4-6, 7-6 (7/5), 6-3.
"I still love competing and still love the game. Obviously it gets harder and harder the older you get," said Murray after clinching his 500th win on hard courts, joining Roger Federer, Novak Djokovic, Andre Agassi and Rafael Nadal in reaching the milestone.
"Yeah, it's not bad (500 hard-court wins). Hard courts have been a great surface for me over the years... I'm very proud of that."
Murray struggled to contain a powerful first-set display from former top-10 player Shapovalov, who is searching for his best form after missing the end of last season through injury.
But the three-time Grand Slam champion edged a second-set tie-break before easing through the decider with a double-break.
Murray will face either fellow veteran Gael Monfils or French fifth seed Ugo Humbert in the last 16.
The 36-year-old Murray has not reached the quarter-finals at an ATP event since his final loss by Daniil Medvedev in Qatar over a year ago.
Murray dropped 17 spots in the ATP rankings earlier on Monday, falling to 67th place.