The 18-year-old, who blasted into the public eye by winning the Indian Wells title in March as a qualifier, won 5-7 6-4 6-4.
Canadian tennis has been on a high with the rise of Andreescu, fellow teenager Felix Auger Aliassime and 21-year-old Denis Shapovalov, but 22nd seed Andreescu is the only one of the trio to reach round two.
Auger Aliassime withdrew injured while Shapovalov lost on Monday. Former Wimbledon runner-up Milos Raonic is injured.
Andreescu missed the claycourt season to rest her injured shoulder, opting to practise in Mallorca.
The lack of matchplay on the dirt courts was apparent in the opening set but she turned things round to grind out victory over 20-year-old Czech Bouzkova in just over three hours.
"I'm not going to lie. This wasn't one of my best matches at all. I was getting pretty mad at myself," she told reporters.
"I mean, it's my first tournament back after an injury, so I tried to stay as calm as possible. And I'm just really proud of how I fought through the match."
Andreescu will play American Sofia Kenin next.
There was no joy for Canada's former golden girl Eugenie Bouchard who went down 6-2 6-2 to Ukraine's Lesia Tsurenko.
(Reporting by Martyn Herman; Editing by Ken Ferris)
