The Red Wings led 3-1 with 13 minutes left in regulation.
Game 4 is Thursday night in Detroit, where the Red Wings will try to stay alive.
The two-time defending Western Conference champions haven't been swept since the first round in 2003 by Anaheim.
Joe Thornton scored 6:42 into the third to pull San Jose within a goal and Logan Couture tied it with 6:43 left.
Tomas Holmstrom and Dan Cleary gave Detroit a 2-0 lead in the first.
San Jose's Devin Setoguchi scored with 4 seconds left in the period, but Henrik Zetterberg restored Detroit's two-goal lead early in the second.