Vasek Pospisil of Vancouver, seeded 28th, played through severe back pain to claim a 3-6, 7-6 (6), 7-6 (9), 6-1 victory over Australian Matthew Ebden.
The Canadian needed treatment three times in the second set and at times looked finished, but the 23-year-old fought on and managed to earn the improbable victory.
The back pain has bothered him since the start of the year, forcing him to quit a semifinal in India and pull out of a tournament in Sydney.
"It's been lingering, I had to hang tough. I'm super-thrilled to get through," said Pospisil, who depended on painkillers to get him to the finish after being told by a physiotherapist that he could go on if he wished.
"Midway through the third I was not feeling very good, I took some tablets and 30 minutes later I didn't feel so much."
The Canadian next plays eighth seed Stanislas Wawrinka after the Swiss beat Alejandro Falla 6-3, 6-3, 6-7 (4-7), 6-4. It will be Pospisil's first career appearance in the third round of a Grand Slam.
Bouchard will play in the third round of a Grand Slam for just the second time after reaching that plateau at Wimbledon last summer. She's the first Canadian woman to reach the third round at Melbourne since Sonya Jeyaseelan in 2000.
The 30th-seeded Bouchard faced a stiff battle against Razzano, who's ranked No. 100 in the world. The Canadian won the first set in 35 minutes but took nearly 90 minutes to win the second.
"It was kind of crazy," Bouchard said of the second set. "It became just mental and I just had to fight. I'm happy I pulled through in the end. We had some crazy points."
"She played some really good points, too, on my match points," Bouchard continued.
"It was a battle back and forth. I felt overall I didn't play as well as I know I can. But even if it's a bit ugly, still fight and try."
Bouchard, 19, is making her main draw debut in Australia after failing to qualify in 2013. She had an early break in the opening set against the Frenchwoman and pulled ahead 5-1.
But she lost a break in the seventh game before eventually taking the set on a Razzano double-fault.
In the second set, the French player fought off match points and took the set to a tiebreaker. Bouchard finally closed it out with a cross-court winner.
Bouchard, who was named The Canadian Press female athlete of the year last month, won with four aces, 25 winners and the same number of unforced errors.
"I came into this Slam seeded now for the first time," Bouchard said. "So, of course, my first few rounds I'm expected to win. I always expect myself to do well.
"I just try to ignore the outside expectations and pressure and really just focus on what I need to do. At the end of the day, I really need to perform. I can't really worry about what people say on the outside."
Bouchard had a handful of Aussie fans in matching singlets spelling out "Eugenie" cheering her on. She threw them her towel after the match and had a koala bear doll tossed back as a memento of the evening.
She next plays Lauren Davis of the U.S., who beat Germany's Julia Goerges 7-5, 2-6, 6-4.
In men's doubles, Toronto's Daniel Nestor and Nenad Zimonjic of Serbia opened with a 6-4, 6-3 defeat of Germans Benjamin Becker and Daniel Brands. The eighth-seeded duo needed just 65 minutes to win with eight aces.
In women's doubles, Sharon Fichman of Toronto and Monica Puig of Puerto Rico dropped a 4-6, 6-4, 7-5 decision in their opening match to Spaniards Garbine Muguruza and Arantxa Parra Santonja.