Bouchard, 19, became the first Canadian woman to reach the round of 16 in Australia since Maureen Drake in 1999. Alexsandra Wozniak of Blainville, Que., got this far at the French Open in 2009 before losing to Serena Williams.
"I thought I played well, tried to be aggressive, really tried to control the point,‘‘ said Bouchard.
Davis, No. 68 on the WTA’s singles list, lost in 70 minutes as both players battled on the fifth day of a record heat wave and temperatures in the mid-40s C.
"It was definitely hot today, the hottest I felt in any of my matches so far,” said Bouchard. ”I was happy to win in straight sets.”
Bouchard, the 30th seed, will next play unseeded Australian Casey Dellacqua, who beat Zheng Jie of China 6-2, 6-4
Bouchard earned her quick win on a second match point, which she set up with a forehand winner in the corner after a 16-shot rally.
The Montrealer took a 5-1 lead in the opening set but failed to serve out in the next game, with Davis climbing to 2-5. Bouchard took the opener in the next game before taking control and running out the victory.
Bouchard ended the match with four aces, breaking on five of eight chances, with 18 winners and 23 unforced errors.
"I always want to do better, I think it's just a step in the process," Bouchard said of her fourth-round achievement.
"I'm trying to always improve on the court. I think I played better today, that's what I'm most happy about."
Dellacqua, unseeded in Melbourne and ranked No. 120 in the world, said she will face a battle against what she called Bouchard’s great server.
”It's going to be another challenge for me ... I know it will be tough against Eugenie."
Bouchard also won her opening doubles match with last-minute partner Vera Dushevina of Russia. They defeated beating Yalena Solovieva and Elena Svitolina of the Ukraine 6-3, 6-1 in a meeting of unseeded pairs.
Milos Raonic of Thornhill, Ont., the 11th seed, will play his third round match on Saturday against No. 22 Grigor Dimitrov of Bulgaria.