Williams, the world number one and top seed, had seized the first set and they were knotted at 1-1 in the second when rain halted play.
The American showed no ill-effects when they returned to the court, wrapping up the victory over the unseeded Australian in 58 minutes.
"I've just been working on some things, working hard," Williams said. "Hopefully they'll come together. I've just been waiting."
Williams stretched her career record over Stosur to 7-3. The Australian stunned Williams in the final of the 2011 US Open, but Williams has now won all three of their meetings since.
Williams still found flaws she wants to improve.
"My toss was a little everywhere. I think I wasn't as relaxed as I wanted to be," Williams said. "But there's always room for improvement Always keep that in mind, that I can continue to improve. So that's one thing that I definitely want to work on."
French Open champion Maria Sharapova had to rally for a three-set victory over Garbine Muguruza to reach the third round.
The fourth-seeded former world number one fought back to beat the rising Spanish talent 4-6, 6-3, 6-1 to set up a meeting with another Spaniard, Carla Suarez Navarro, who was a 6-1, 3-6, 6-3 winner over Czech Karolina Pliskova.
Sharapova, playing her first match since a surprise fourth-round exit at Wimbledon, was a runner-up at this event in 2009 when it was held in Toronto.
She was pleased to get her campaign off to a winning start against a player who won her first career singles title at Hobart this year and has four career wins over top-10 players on her resume.
"It's been a few weeks since I've been in that competitive atmosphere," Sharapova noted. "It's always quite different, no matter how much you train, you try to prepare for that.
"Once you step on the court, you feel a little bit more from the crowd, the energy. You get inspired by everything. But you're ultimately a bit rusty.
"I was just happy that I got through that and won today because I had a very tough opponent."
Sharapova had rallied from a set and 5-4 down to beat Muguruza in the quarter-finals of the French Open.
This time, too, Muguruza gained the early momentum, going up a set and a break before the Russian superstar stormed back, winning four of five games to force a third set that she dominated.
"I was happy that I was able to turn it around," Sharapova said. "I didn't feel good. I made a lot of errors in the first set. When you're able to finish stronger than your start, that's always a positive because you give yourself an opportunity to keep playing in the tournament."
- Serena seeks 18th Slam -
Meanwhile, sixth-seeded German left-hander Angelique Kerber made short work of France's Caroline Garcia 6-4, 6-1 and 11th-seeded former world number one Caroline Wozniacki steamrolled Czech Klara Koukalova 6-1, 6-2.
Kerber has reached, and lost, four finals this season, including losing to Williams in the final at Stanford on Sunday.
Williams' title in Stanford was her fourth of 2014, but her performance in the year's Grand Slams has been disappointing.
The 17-time major winner hasn't made it past the fourth round in the Australian Open, French Open or Wimbledon, and the US Open starting August 25 will be her last chance of the year to add another Slam to her resume.
She next faces 15th-seeded Czech Lucie Safarova, a 6-4, 6-2 winner over Slovakian Magdalena Rybarikova. Williams has won all six meetings with Safarova.
"She is an interesting player. She's extremely tricky. I don't know how I got to be 6-0 against her because she plays really well," Williams said.
"It's definitely not an easy match for me. I look forward to it.I'm going to have to be ready. She's so motivated and she's playing really well."
Second-seeded Czech Petra Kvitova, who claimed her second Wimbledon title last month, defeated Aussie Casey Dellacqua 6-3, 6-2.