The 29-year-old second seed, who won the tournament in 2009 in his last appearance, will now be the favourite after top-seeded American Jack Sock was bundled out by unheralded German Peter Gojowczyk.
Del Potro will next face Russian Karen Khachanov, who beat sixth-seeded Uruguayan Pablo Cuevas 6-2 7-6(4), in the last eight.
"I was surprised by the match today," said Del Potro, who had lost to Shapovalov in Montreal last August. "I didn't expect to play at this level in the first match of the year.
"But I've been working hard to play this tennis and hopefully I can still improve in the next matches.
"If he has a good day, he is very dangerous as a lefty, he serves well and volleys. Denis played better than me in the Montreal match, but today I broke him three times to love and that was the key."
Defending champion Sock lost 6-3 6-3 in only 61 minutes to his German opponent, who is ranked 65th in the world.
The Australian Open, the first grand slam of the year, runs from Jan. 15-28.
(Reporting by Sudipto Ganguly in Mumbai; editing by Peter Rutherford)
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