''I just tried to change the angle, slow down and make him move a little bit,'' Vermette said of his game winner. ''I just made a move and got it between his legs.''
Mikkel Boedker also scored for the Coyotes (10-3-2), who have points in 11 of their last 12 games.
Smith had 48 saves in regulation and overtime, and then stopped four of the five shots in the shootout.
''He was big, especially in the power play,'' Vermette said. ''He made some key saves and was an important part of our win tonight.''
Joe Pavelski and Danny Boyle, who missed the previous seven games with a concussion, scored for the Sharks (10-1-3).
Logan Couture converted the first attempt of the shootout for San Jose. He earlier thought he had the winning goal with 33 seconds left in regulation. However, it was waved off and replays clearly showed the puck never crossed the line.
It was just one of many brilliant saves made Smith. Sharks goalie Antti Niemi shopped 28 shots.
''We had a lot of real good scoring chances,'' Sharks captain Joe Thornton. ''It felt like we were in their zone a lot.''
Radim Vrbata also scored in the shootout for Phoenix.
''Mike Smith was huge in this game,'' Coyotes' coach Dave Tippett said. ''He made save after save and we managed to get points in the shootout.''
Phoenix peppered Niemi with seven shots during the 5-minute overtime, with Niemi coming up big. The Sharks had some chances but did not get a shot on goal.
The Coyotes scored just 36 seconds into the game. Rob Klinkhammer picked up a loose puck along the boards near the blue line and fed Vermette, who beat Niemi to the far post.
''It was good work on the forecheck,'' Vermette said. ''We forced the turnover and found a way. I just shoot it and it found a hole.''
Pavelski's goal came just over a minute into the second period, tying the score at 1. Tommy Wingels won a scrum in the Sharks' zone and sent it up the ice to Matthew Nieto, who missed the previous two games with an injury.
Pavelski took the pass and skated in, looked off Smith and found an opening to the near side.
''We put ourselves in a hole with the penalties we took,'' Tippett said. ''There were too many turnovers that gave them too many chances.''
The Sharks took a 2-1 lead with a power-play goal four minutes later. Pavelski's shot was blocked and the Coyotes couldn't clear it and Thornton kicked it back out to Pavelski, who found Boyle on the left side. Boyle put it into the upper far corner.
''That definitely felt good,'' said Boyle, who slammed his hands into the glass behind the net as part of his celebration. ''I just felt a half-second behind everything. I did not feel like my normal self but I will get better in a hurry.''
Boedker took a pass from Derek Morris midway through the second period and slipped it along the ice to the far side to tie the score at 2.
NOTES: Sharks C Joe Thornton recorded his 800th career assist on Boyle's goal. ... Coyotes C Mike Ribeiro played in his 800th career game. ... Boedker ended a three-game scoreless streak. ... Vermette has scored goals in consecutive games. He did not record a point in his previous four. ... Morris has recorded points in eight of his last 11 games. ... The Coyotes are outscoring their opponents 22-15 in the second period. ... Pavelski has scored goals in three straight. ... The Sharks have taken 50 or more shots in both home games against the Coyotes this year. ... Vrbata improved to 32 of 73 on penalty shots.