Wozniak, ranked 55th to Hantuchova's 35th, broke serve for a 6-5 second-set lead, but let three match points slip away to take it to a tiebreaker in the hard-court event at Uniprix Stadium. Then a light rain wet the court just enough to force a 55-minute delay.
The players returned to mostly empty seats at center court.
"I tried to finish it when I had the match points, but it wasn't easy," Wozniak said. "I gave everything on every point. It was important for me to win here."
Wozniak, who has not advanced beyond the second round in nine previous Rogers Cups, will next play former world No. 1 Jelena Jankovic. The Serb is 4-0 against Wozniak, winning once in 2008 in Montreal.
Vinci, ranked 28th, trailed 5-2 in the third set when she began winning points on her forehand against the Belgian.
She advanced to play 11th-seeded Ana Ivanovic, the 2006 Rogers Cup winner.
The tournament began a day later than usual to accommodate the London Olympics.
In other matches, Chanelle Scheepers of South Africa rallied to beat Stephanie Dubois of Canada, 1-6, 6-3, 6-0; Ekaterina Makarova of Russia handled Urszula Radwanska of Poland 6-4, 6-2; and Tamira Paszek of Austria defeated Julia Goerges of Germany 6-2, 6-1 to earn a matchup with top-seeded Victoria Azarenka.
Azarenka is hanging onto the No. 1 world ranking by a thread and will have to hold off second-seeded Agnieszka Radwanska and No. 3 Maria Sharapova to stay on top. Sharapova can regain the top ranking if she wins the tournament, while Radwanska would need to win and have both Azarenka and Sharapova bow out before the quarterfinals.
"It's really exciting," said Azarenka, who captured her first grand slam title this year at the Australian Open. "We're all showing consistent tennis through the year."