Montreal's Bouchard survived a scare from a 431st-ranked challenger from China, scoring a 7-5, 6-1 victory over Chen Tang Hao.
Vancouver's Pospisil, who is seeded 28th, eliminated Australian Sam Groth with a 6-4, 6-3, 6-4 win.
Bouchard, the 30th seed, rode a roller-coaster in the one-hour opening set after earning a 3-1 lead, only to lose it a game later on a double-fault. Bouchard, recently named The Canadian Press female athlete of the year, was broken again to drop to 3-4 as Tang rifled back a return winner before holding for a 5-3 lead.
Bouchard saved set points in the ensuing game on Tang errors before finally earning a 7-5 win on a break.
The 19-year-old Bouchard was more in control in the second set and dominated to win the 6-1.
"It was tough but I fought and got through it," Bouchard said. "I didn't play the way I really wanted, but it was good to win one like this.
"It was really fun, I had some good support, it was good atmosphere," said Bouchard, who admitted struggling the 35-degree heat.
"It was hotter than past few days, tough for both of us," she said. "It's strange training at one temperature (around 25 C on Sunday) and playing in something completely different. But no complaints, it was the same for both of us."
She said she needed to be mindful of the hot sun.
"I'm big on sunscreen and hydration, especially the night before," said Bouchard. I did a good job of it and was fine. I'd rather play tennis in the sun than do anything else anyway."
Pospisil, 23, was never troubled by Groth and appears to have overcome a back problem which bothered him this month in India and Sydney.
In the second round, Pospisil will face another Australian in Matthew Ebden, who beat Nicolas Mahut of France 6-3, 7-5, 4-6, 0-6, 6-3.
Pospisil said that while he did not feel at 100 per cent, he played well.
"I played surprising well, I was not expecting to play like this," he said. "I wasn't feeling that great and only played points the day before, I didn't have much preparation.
"But I returned and served well, overall I did fine."
"I didn't feel any fatigue, I was playing relaxed because of my back," he added. "I don't anticipate any problems in the next round but I'm glad I didn't have to go five sets."
Pospisil overcame 16 aces from his opponent, firing six of his own and breaking on four of 10 chances. He scored 31 winners and 17 unforced errors.
The Canadian earned the break he needed in the seventh game of the opening set and in the final game of the match on the Groth double-fault.
Note to readers: This is a corrected story. An earlier version reported that Pospisil was making his Grand Slam debut.