Second-seeded Sharapova was broken once in the first set before she won the last eight games of her 6-3, 6-0 fourth-round victory over No. 21Peng Shuai.
Seventh-seeded Bouchard, who reached the semifinals or better at the first three Grand Slam tournaments last year, won nine of the first 10 games against Irina-Camelia Begu but lost seven of the next nine to be pushed to a third set for the first time in the tournament.
After serving a double-fault on set point to end the second, Bouchard took a short break before returning to complete a 6-1, 5-7, 6-2 win over No. 42 Begu, who had taken out nine-seedAngelique Kerber in the first round.
"I gave myself a good, long hard look in the mirror," Bouchard said of her brief absence from Rod Laver Arena. "I said, 'Genie, this is unacceptable.' I really kind of kicked myself in the butt a little bit."
After three straight-sets wins, Bouchard joked she went three sets for a couple reasons.
"Clearly, I need more practice!" she said, then turned to the section of fans known as the Genie Army, who support her in Australia. "It's not horrible playing longer on this court. I just wanted to do that for you guys and for them to practice more songs."
Five-time Grand Slam winner Sharapova has a 3-0 career record against Bouchard, including a comeback semifinal win last year at the French Open. She had a rougher road to the quarterfinals in Melbourne Park, though, and had to save match points in her second-round win over Russian qualifier Alexandra Panova. Sharapova lost in the fourth round last year at Melbourne Park, while Bouchard reached the semifinals in her tournament debut.
"I feel like something or someone gave me another chance," Sharapova said. "Last year, I lost in the fourth round here, [so] getting to the quarters is really special."
Sharapova said Bouchard was the most consistent player in the 2014 Grand Slams, and she had to be at her best to beat the 20-year-old Canadian.
"She's playing really well, confident tennis, so aggressive," Sharapova said. "I have a tough match ahead of me, but I always look forward to that."
In the other quarter of the draw, No. 10 Ekaterina Makarova had a 6-3, 6-2 win over Julia Goerges to reach the last eight, where she'll play the winner of a Sunday night match between No. 3 Simona Halep and Yanina Wickmayer.