The seventh seed from Westmount, Que., needed just 54 minutes to defeat the 72nd-ranked Bertens in their first career meeting.
Bouchard rolled through the opening set in just 23 minutes against an opponent who is now 0-5 versus top-10 opponents.
Two of Bertens' three game wins came on breaks of Bouchard's serve, including when the Canadian was first serving for the match.
Bouchard finished with six breaks of serve and 20 winners as she won a second consecutive match on the Margaret Court showcase arena.
"I think I put pressure on her a lot and I think probably forced her to make some errors," said Bouchard. "I was happy with the way I was able to step in and control the points, whether I did the winner or she did the error."
"In the end I think that's how I want to play. Just need to stay a bit more focused at the end to try to close it out. More on my terms."
Bertens briefly made a match of it in the second set, but fell behind despite breaking Bouchard in the opening game of the second set and losing the lead a game later.
"I lost my focus a bit, that's unacceptable," said the 20-year-old. "I'll make sure it won't happen again. I can be hard on myself, I'm a perfectionist."
"It's fun to play here at night, it's a special atmosphere," added Bouchard. "I'm glad I was finally able to hold my serve — and hold my serve at the end."
Bouchard, who broke through to the semifinals in Melbourne and Paris last season before playing in the Wimbledon final, is now 7-1 in Australia.
Bouchard will play in the third round against France's Caroline Garcia, who beat Swiss player Stefanie Voegele 6-3, 6-4, coached by her own former mentor Nathalie Tauziat.
"I worked with Nathalie a little bit, I guess that's how the tennis world is," said the Canadian. "It's a small world.
"It's kind of musical chairs in terms of coaching. I'm sure it won't be the last time in my career. It won't really bother me. I'm not playing against the coach, I'm playing against the player.
"I'm not going to worry about it. I'm still friendly with Nathalie, so it's all good."
In men's doubles, the 13th-seeded duo of Vancouver's Vasek Pospisil and Austria's Julian Knowle opened with a 7-6 (3), 3-6, 7-6 (4) victory over Germany's Jan-Lennard Struff and Dominic Thiem of Austria.
Pospisil won the Wimbledon doubles title last summer with American Jack Sock, who is still recovering from hip surgery.
Toronto's Daniel Nestor is playing with India's Rohan Bopanna this season. The seventh-seeded duo will open against Marcos Baghdatis of Cyprus and Marinko Matosevic of Australia.
Nestor and Bopanna won their first title together last weekend in Sydney.