The Canadians clinched the Group A match against the United States when Vancouver's Vasek Pospisil beat John Isner 6-3, 7-6 (4) in the men's singles.
Williams asked for and was served a coffee after losing her first set of the tournament 6-0 to Flavia Pennetta on Monday, and credited the espresso with her rapid return to form and comfortable win that set the U.S. on course for a 3-0 victory over Italy. There was no such comeback against Bouchard in a match that lasted only 50 minutes.
"Maybe she needed another coffee," said Bouchard, from Westmount, Que., who had never previously beaten Williams. "I know she didn't play her best. I'm happy I just stayed with it and held my nerve. It's a good way to start the year — gives me some confidence."
Williams, who arrived in Perth in Saturday, again appeared lethargic and later complained of fatigue.
"It's weird. I can't get my body to move. I feel like I've got no energy," Williams said. "It's a little frustrating because I know I can play 2,000 times better.
"I've just got to get my feet moving. I have to figure it out."
The Czech Republic beat Italy 3-0 in the other Hopman Cup match.
Elsewhere at the Brisbane International, second-ranked Maria Sharapova opened her 2015 season by winning nine straight games in a 6-0, 6-1 win over Yaroslava Shvedova.
The Russian had a bye in the first round at the season-opening WTA event in the sub-tropical east coast city, where Williams won the title in 2014 but skipped this year in favour of the Hopman Cup.
Sharapova needed little more than an hour to oust Shvedova, a qualifier from Kazakhstan. Sharapova wrapped up the first set in 23 minutes and led 3-0 in the second before Shvedova held serve, raising her arm in mock triumph. Shvedova had a break-point opportunity in the next game, but couldn't put away an overhead and Sharapova responded with a backhand winner down the line.
"It certainly felt good to start, after not playing a match for a couple of months," said Sharapova, who is into the quarter-finals in her first competitive tournament since the season-ending WTA Finals in Singapore in October. "I wanted to start off sharp and try to do the right things ... and I think I did a good job of that.
"There were moments where I saved a few important break points, which was crucial. That gave me good confidence."
Third-seeded Angelique Kerber advanced earlier Tuesday with a 6-3, 7-5 win over Daria Gavrilova. Varvara Lepchenko progressed with a 6-4, 6-4 win over fellow American Madison Keys.
In a men's first-round upsets, Lukasz Kubot of Poland beat fifth-seeded Kevin Anderson of South Africa 7-6 (3), 6-4 and Sam Groth beat defending champion Lleyton Hewitt 6-3, 6-2 in a night match. Seventh-seeded Alexandr Dolgopolov of Ukraine advanced over Carlos Berlocq of Argentina 6-2, 6-3.
At the ASB Classic in Auckland, New Zealand, third-seeded Venus Williams made short work of her first match of the new year by sweeping past Jana Cepelova of Slovakia 6-4, 6-0 in 52 minutes.
The seven-time major winner reached the final of the Auckland hard-court tournament last year.
"I was welcomed when I came on the court and I felt like I left off right from last year," Williams said. "I felt comfortable right away."
Top-seeded Caroline Wozniacki had to work hard to subdue qualifier Julia Glushko of Israel 6-3, 6-2.
Urzula Radwanska beat former French Open champion Francesca Schiavone 6-4, 7-5 and another qualifier, Lucie Hradecka of the Czech Republic, toppled fifth-seeded Svetlana Kuznetsova 3-6, 7-6 (6), 6-4