San Jose must defend home ice Thursday night to avoid becoming the fourth NHL team to lose a best-of-seven series after leading 3-0. San Jose's Logan Couture scored early in the third period, but Zetterberg tied it with 9:22 remaining, and Filppula scored the winner with 7:28 left.
Darren Helm added an empty-net goal. Every other game in the series was decided by one goal.
Detroit scored the last three goals of Game 5 to win 4-3.