Phoenix had lost eight straight to San Jose and limped toward the end of the season, putting its playoff chances in doubt. The Coyotes seemed to be rolling toward ending both dilemmas, building a three-goal lead early in the third period.
San Jose came charging back on Joe Thornton's goal for his 1,000th career point and Danny Heatley's 26th goal of the season with 3:39 left.
Bryzgalov finished it off to become Phoenix's career wins leader, sending the Coyotes to the playoffs in consecutive seasons for the first time since 1999-2000.
Ray Whitney and Andrew Ebbett also scored, and Radim Vrbata had a goal in his 600th career game for Phoenix.
Ian White had a first-period goal for San Jose, which missed out on a chance to lock up No. 2 seed in the West after Detroit lost to Chicago.
Phoenix seemed to be in prime position to lock up a playoff spot after a five-game winning streak in March. The Coyotes hadn't been able to finish it off, though, losing three of their previous four games, the last two in shootouts.
That set up a make-or-break weekend, where the possibilities ranged from fourth in the Western Conference with a home-ice advantage to ninth and watching the postseason at home. Phoenix needed to either win one game or have Dallas lose just one, but the Stars weren't cooperating; they beat Colorado for the second time in two nights earlier Friday.
Worse yet, the Coyotes were facing San Jose, which had already locked up the Pacific Division title and outscored them 17-6 in four games this season.
Phoenix came aggressively to knock the Sharks on their heels and it paid off late in the first period, when hard hit by Paul Bissonette set up Ebbett's second of the season on pass from Kyle Turris that bounced off his skate.
White tied it a couple minutes later with a shot that caromed off someone past Bryzgalov, but the Coyotes seemed to take control after that.
Whitney put Phoenix back up with a one-timer from Doan, Vrbata followed with a 2-on-1 one-timer from Eric Belanger 2½ minutes later and Doan became the Coyotes' first 20-goal scorer this season early in the third.
The Sharks roared back in the final 9½ minutes, getting goals from Thornton and Heatley and continuing to put pressure on Bryzgalov. Phoenix's goalie closed it out with several tough stops through traffic, including three in the final minute, to pass Bob Essensa's team record with his 130th career win.
NOTES: San Jose C Scott Nichol returned after missing 19 games with an upper-body injury, but RW Devin Setoguchi didn't play after sustaining an undisclosed injury during warmups. ... Phoenix C Martin Hanzal returned after missing 14 games with a lower-body injury.