Hurkacz won 2-6, 7-6 (2), 3-6, 6-3, 6-3. The players were on serve in the fourth set when the match was suspended Monday night because of rain.
The versatile Hurkacz, 24, won 50 points at the net, including 10 when playing serve-and-volley. He became a first-time Grand Slam quarterfinalist, and his opponent Wednesday will be his idol, Roger Federer.
Federer, 39, became the oldest man to reach the Wimbledon quarterfinals in the Open era with his 105th career victory at the tournament on Monday.
"Roger, what he does, the way he plays, the titles he has won -- he has inspired so many people," Hurkacz said. "It's going to be fun. I'll be hoping to get a little bit of support."
The Associated Press contributed to this report.