Attacking Del Potro's serve at 4-4 in the third, Basilashvili earned four chances to break and serve for the match. But the 2013 finalist defended resolutely and switched gears to set up a meeting with Russian teenager Andrey Rublev.
South Korea's Chung Hyeon eased past world number 13 Roberto Bautista Agut, last year's runner-up, 6-4 6-3 to equal the best victory of his career.
World number 60 Chung, who beat David Goffin when the Belgian was ranked 13th in Montreal this year, next faces Richard Gasquet.
American qualifier Frances Tiafoe stunned Benoit Paire 6-4 6-4 and fired a warning to the experienced players on the Tour.
"A lot of us younger guys are playing well, doing well, and it's going to help others step up," he said. "I think you're going to see a lot of us in the top 50. The other guys are going to have to leave eventually."
Also advancing to the next round were Feliciano Lopez who defeated big-serving Ivo Karlovic 7-6(1) 7-6(6) and Britain's Kyle Edmund who downed Jiri Vesely 6-3 6-2.
Roger Federer, the 2014 champion, starts his campaign on Wednesday.
"I couldn't be happier with the season, to be quite honest," Federer told reporters.
"I'm relieved that I'm feeling better since the American summer and Canada. It would be nice to pick up one more title, maybe a couple more would be nice, and then get into training and start again next year."
The Swiss 19-times grand slam winner begins his campaign against the winner of Tuesday's clash between Diego Schwartzman and Jordan Thompson.
(Reporting by Shrivathsa Sridhar in Bengaluru, editing by Ed Osmond)
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