Murray struggled to find his usual rhythm in windy conditions and had to save two set points in a mixed display that reflected some nerves and a lack of match practice.
The Scot romped through the tiebreak with some impressive shot-making to set up a third round meeting with Latvia's Ernests Gulbis, who beat Italian Fabio Fognini 6-3 1-6 6-1.
"It was tricky, it is extremely windy here," Murray said in a courtside interview. "I didn't feel that I had my timing and got pushed behind the baseline."
Nadal was more impressive in his first singles match since a shock first round defeat by Steve Darcis at Wimbledon opened the door for Murray to win the title.
The 12-times grand slam champion found Levine willing in the opening exchanges as the pair engaged in some exciting and lengthy rallies.
But Nadal, as he has demonstrated throughout his illustrious career, had all the answers and showed in his 6-2 6-0 victory that he has returned to the tour in top shape after an absence of six weeks.
"After seven weeks without playing a match, I'm very pleased," Nadal told reporters.
"This is very tough tournament, a Masters 1000 in Montreal, on a fast court, so to be through in the third round ... I have to be proud and happy about the way that I played."
Canadian wildcard Vasek Pospisil proved his upset of world number 20 John Isner was no fluke as he stormed to a 6-2 6-4 victory over Czech Radek Stepanek.
Pospisil, ranked 71st in the world but playing well above that lowly standing, will now face fifth seed Tomas Berdych who wobbled at times in his match against Alexandr Dolgopolov but emerged a straight sets winner.
Teenaged Canadian Filip Peliwo missed out on a dream match against Novak Djokovic, however, losing in three sets to Denis Istomin who will face the Serbian world number one next.
German Tommy Haas, however, lasted just 19 minutes before he retired trailing 5-0 to Australian qualifier Marinko Matosevic.
Frenchman Bernard Paire, who overcame Swiss Stanislas Wawrinka, awaits Matosevic in the last 16.
Later on Wednesday, Argentine sixth seed Juan Martin Del Potro, fresh off winning last week's Washington title, meets Croatia's Ivan Dodig.
(Editing by Martyn Herman)
