David Moss had two goals, and Vrbata scored late in the third period to help send Phoenix to overtime for the fourth straight game.
The Coyotes were much better in their own end against San Jose, allowing 28 shots after giving up 101 combined in a pair of games in San Jose earlier this season.
They still lost for the fifth time in six games after struggling in the shootout.
San Jose had to fly out Friday morning because the new NHL labour agreement requires teams to take three full days off for the holiday break.
The Sharks seemed a step slow in the opening period, and the Coyotes jumped on them, generating several good scoring chances.
San Jose gave them an extra one midway through the period, when Sharks forward Matt Nieto flipped an errant pass right to Moss, who whirled around in the slot and beat Niemi.
The Sharks nearly scored early in the second period when a shot by Justin Braun hit the left post, bounced off a stick and hit the right post before being cleared away.
They did get one through after a Coyotes turnover. Thornton ended up with the stray puck, skated up the middle and found Pavelski, who beat Smith over the glove shoulder from the left circle.
The next two goals came quickly.
Phoenix answered 68 seconds after Pavelski's goal when Moss fought off Nieto and redirected a shot from David Schlemko past Niemi.
San Jose responded on a power play two minutes after that, tying it when Marleau gathered a pass from Tommy Wingels and beat Smith with a wrist shot from between two defenders.
The Sharks made a big push to start the third period, and Burns scored midway through, one-timing a feed from Pavelski behind the net that went through Phoenix defenceman Derek Morris' legs.
Phoenix again had an answer, scoring with just over 6 minutes left on a power play when Mike Ribeiro sent a pass through traffic to Vrbata, who scored his 11th of the season from the left circle to tie it at 3-all.
NOTES: Friday night was the 10th anniversary of the Coyotes' first game at Jobing.com Arena — a 3-1 loss to Nashville. ... Thornton, the NHL assists leader with 37, has five in his past two games. ... Coyotes captain Shane Doan missed his ninth straight game with Rocky Mountain Fever. He skated Friday morning, but no timetable has been set for his return. ... Couture has gone seven games without a point, leaving him stuck on 99 goals and 98 assists for his career.