Williams spoke after her match but before she knew who she would face in the quarter-finals, saying either Osaka, who could return to the world number one ranking next week or Swiatek would be worthy foes.
Bouzkova's week included wins over 2017 U.S. Open champion Sloane Stephens, former French Open victor Jelena Ostapenko and a clash with Wimbledon champion Simona Halep, who retired hurt after dropping the first set of their quarter-final.
There was no such drama in Toronto, where Williams broke Osaka for a 5-3 lead in the first and closed out the opening set with a service victor. "And, you know, I'm just overall a little bit better".
Third time was the charm for Serena Williams against Naomi Osaka.
Williams had slow starts in her previous two outings this week but came out much stronger for this one and delivered a serving masterclass that denied Osaka any break point opportunities.
Andreescu shot to prominence with a victory at Indian Wells in March, but this week marks her first tournament since she was forced out of the French Open with a shoulder injury.
That would prove to be crucial as she broke in the following game, taking a 3-1 lead she would not relinquish.
A win for Williams sees her bounce-back after losing in disastrous conditions at the Arthur Ashe Stadium in NY, when the 37-year-old received code violations for coaching, smashing her racket, and calling the umpire Carlos Ramos a "thief" at the 2018 US Open. Japan's World Number one tennis player felt the same as she used to feel a few months ago when she won in Melbourne.
Meanwhile, Osaka, 21, explained that Williams "is someone that I looked up to forever".
"I'm clearly not (yet) where I would like to be but I'm getting close to it", responded Williams, still looking for his 24th Grand Slam, which would be the first since the birth of his daughter in September 2017, and who would allow him to match Margaret Court's record.
Youngsters Bianca Andreescu and Sofia Kenin will contest the other semi-final.
Williams will face the victor of the quarter-final match between Simona Halep and Marie Bouzková, in the last four.
"In the third set I told myself, 'Go big or go home, ' " Andreescu said.
Williams is no stranger to the Rogers Cup. I feel good still.