Bouchard, ranked 18th in the world, hit 36 winners and needed just 75 minutes to dispose of 10th-ranked Errani 6-3, 6-3.
The 20-year-old Bouchard won 76 per cent of her first serve points in the match. Her next match will be against No. 7 Simona Halep of Romania.
On the men's side, Vancouver's Vasek Pospisil fell 6-0, 6-2 to Mikhail Kukushkin of Kazakhstan in the second round in just over an hour. It was Pospisil's first match after getting a first round bye as the tournament's No. 25 seed.
Spain's Roberto Bautista Agut knocked out Tomas Berdych, also in the second round, beating the fourth-seeded Czech player 4-6, 6-2, 6-4.
Berdych had won 11 matches in a row in February and is 16-4 this year.
"He really handled the tough situation today really well," Berdych said. "But secondly, more important to me, anything I touched today was basically bad and was wrong. So really, just to recap it, definitely my worst match that I had this year. That's it, you know."
Two-time tournament champion Novak Djokovic, seeded second, was a 7-6 (1), 6-2 winner over Victor Hanescu of Romania in their second-round match.
"I was not very happy with my performance but it was a straight set win," Djokovic said. "I'm going to have to work very hard on the practice court to get better."
Second-seeded Agnieszka Radwanska coasted into the fourth round with a 6-0, 6-0 victory over Annika Beck of Germany, and 2011 champion Caroline Wozniacki topped qualifier Yaroslava Shvedova of Kazakhstan 7-6 (7), 3-6, 6-1. In the late match, No. 7 seed Jelena Jankovic overcame back problems to defeat Magdalena Rybarikova of Slovakia 6-3, 6-2.
Sixth-seeded Juan Martin del Potro withdrew because of ligament damage in his left wrist. The Argentine player also pulled out of the Dubai tournament last week.
Ninth-seeded Jo-Wilfried Tsonga of France also dropped out, falling 6-4, 6-4 to countryman Julien Benneteau.
John Isner, the No. 12 seed and the only remaining American from the original 14-man contingent, advanced with a 7-6 (5), 6-3 victory over Russia's Nikolay Davydenko. Isner, slowed by an ankle injury earlier this season, said he feels good physically and mentally.
"It's a situation that's very good for me," the 6-foot-10 former University of Georgia player said. "In the U.S., (on a) hard court that bounces high and pretty slow, and have the fans on my side, I can't ask for anything more. All the surroundings bode extremely well for me. So I'm in a good place."
He said he wasn't concerned about upholding American honour, either.
"No, I don't," he said. "I guess yesterday wasn't a great day for our American men. But, you know, selfishly, I'm just worried about myself, really. I certainly want all those guys to do well. But at the same time, it's not my concern at all. My concern is myself and just trying to keep moving forward and keep getting better and get back to where I want to be."