Anaheim was coming off consecutive shootout losses to Los Angeles and San Jose. It had lost four tiebreaker games in a row since it beat Phoenix 3-2 on Oct. 18 in its first shootout game of the season.
"It's been a while," said Hiller, who had 23 saves in regulation and overtime.
After Jonathan Toews scored for Chicago on the first shootout attempt, Nick Bonino responded for Anaheim. Hiller then stopped Sharp, and Kyle Palmieri beat Antti Raanta with a backhand to give the Ducks the advantage.
When Hiller made the save on Kane at the other end, it was over.
"Once the Bonino, the fluky one, went in, it was, 'OK, the monkey's off our back. We're good to go. We're going to start scoring again,' because we missed about 18 in a row," Ducks coach Bruce Boudreau said.
Raanta had 24 saves in his first home game. It was the first NHL shootout for the rookie, who won each of his first two starts. He looked as if he could have stopped Bonino's attempt in the tiebreaker, but it trickled under him for the score.
"Now I know how the shootout's going to be in the NHL, so we'll have to keep on working with those also," Raanta said.
It's the first three-game losing streak for the Blackhawks during the regular season since February 2012. They also lost three in a row against Detroit in the Western Conference playoffs last season but managed to advance in seven games on their way to the title.
"Frustrating. You can't overreact to it," Toews said of the slide. "You just have to keep improving on the little things we do well as a team, and know that if (we) keep working, we'll eventually get the breaks."
"We continue to know how to play against these guys," Perry said. "We have a lot of great teams in the West, and they're one of them."
Getzlaf snapped a 1-all tie at 10:05 of the second, using a screen by Perry to send a slap shot through Raanta's legs. Getzlaf's 15th goal came with Johnny Oduya in the penalty box for closing his hand on the puck.
Anaheim remained in front until Bryan Allen was sent off for boarding Toews. The Blackhawks made the most of the power-play opportunity, with Versteeg beating Hiller into the upper right corner with 2:12 to go in the second.
Versteeg's second goal since he was reacquired in a trade with Florida on Nov. 14 stopped a streak of 26 consecutive kills for the Ducks.
Each team had ample opportunities to move in front in the third, but Hiller and Raanta took turns turning away prime chances. Hiller denied a streaking Sharp right in front of the goal, and Raanta grabbed a slap shot by Matt Beleskey.
Hiller also made a terrific kick save on Toews about 35 seconds into overtime, and the Ducks had a potential winning goal waved off.
"That was a tight game," Blackhawks coach Joel Quenneville said. "Had a couple good chances late in the game that were dangerous and a couple big stops."
Getzlaf leads the Ducks with 19 assists. He has at least one point in each of his past 12 games. ... Blackhawks C Michal Handzus returned to the ice after missing two games with an illness. C Andrew Shaw (upper body) could return as soon as Sunday against Florida, while LW Bryan Bickell (lower body) is expected to begin skating again in the next week. ... The Ducks scratched forwards Tim Jackman and Patrick Maroon and defenseman Alex Grant. ... The Blackhawks' charity organization launched an online raffle for a chance to win a 2013 Stanley Cup championship ring.