Fourth seed Wozniak wrapped up the first set without much fuss although Vesnina, who was ranked as high as 22 in the WTA rankings in 2009 but is now at 81, hit back.
The Canadian held her nerve, however, and completed a 6-2 4-6 6-3 win over her Russian opponent to the delight of the Canadian crowd.
Czech sixth seed Barbora Zahlavova Strycova had to battle through a tough first set before eventually powering past Marie-Eve Pelletier of Canada 7-6 (7/2) 6-1.
Petra Martic and Lucie Hradecka, seeded seventh and eighth respectively, were also given stiff workouts.
Croatian Martic sealed a 6-4 7-5 win over lucky loser Jessica Pegula from the United States while Czech Hradecka claimed a 6-4 7-6 (7/5) victory over Maria Sanchez.
Wild card Eugenie Bouchard thrilled her home fans by coasting past the USA's Grace Min 6-4 6-3.
Kristina Mladenovic and Lauren Davis, both of whom qualified for this tournament, advanced after seeing off Madison Brengle and Stephanie Dubois respectively.
There were also wins for Anna Tatishvili, Mirjana Lucic, Melanie Oudin and Petra Rampre.