In the first match on centre court, Azarenka and Magda Linette twice had play halted before the 16th seeded Belarusian finally prevailed 6-3 6-0 in a contest that took nearly five hours to complete.
Azarenka was unfazed by the rain, charging ahead 5-2 in the opening set before closing it out with a break just before play was stopped a second time.
Returning to the court three hours later, Azarenka was taking no chances of another delay. She romped through the second set, winning 24 of 31 points and breaking Poland's Linette three times.
Azarenka now awaits the winner between American Sloane Stephens and Ukraine's Anhelina Kalinina.
Pliskova (31) wobbled out of a second set rain delay losing the tie-break but recovered to dispose of China's Zhu Lin (29) 6-3 6-7 (8) 6-2 to register just her second match win in her last six events going back to April in Stuttgart.
Pliskova, who parted ways with coach Sascha Bajin two weeks ago, easily took the opening set and had looked headed for a straight-forward victory before Zhu fought off two match points and rain halted play with the tie-break deadlocked at 8-8.
When play restarted, Zhu pounced, winning two points to clinch the tie-break 10-8.
But the Czech quickly refocused in the decider, applying pressure and breaking Zhu three times to move onto the second round and a meeting with Polish world number one and reigning U.S. Open champion Iga Swiatek.
"I was kind of ready for both situations because it's 8-8, so it can just be like two quick points, so you have to be ready that you are going to lose them as well," Pliskova said.
"Of course, it's tough because you kind of are so close to the win, and then you just start from zero going in the third set, but there's nothing you can do. You have to still be there and fight."
Swiatek arrives in Canada riding the momentum from a victory at her home tournament the Warsaw Open, but fighting to fend off a challenge from Aryna Sabalenka, who can take over the top spot in the rankings with a win in Montreal and if the Pole exits before the quarterfinals.
In the evening session, American Madison Keys (28) got the better of her veteran compatriot Venus Williams (43) 6-2 7-5.
Williams stepped up her game in the second set to prevent Keys from crossing the finish line, saving seven match points before breaking serve to the delight of the crowd.
But Keys broke right back and after a visit from a physiotherapist, sealed the win in the next game to set up a second-round meeting with Italy's Jasmine Paolini.
Spaniard Paula Badosa announced her withdrawal from the tournament on the eve of her first-round match due to a back injury.
