Dominik Hasek made 31 saves and turned the Ducks away during a late flurry that saw them put three shots on goal in the final 24 seconds.
The other Detroit goal came from Henrik Zetterberg on the power play just 3:34 into the game. Nicklas Lidstrom assisted on Holmstrom's goal in what was Detroit's first conference finals appearance in five years.
Anaheim's goal was scored by Chris Kunitz early in the second period.
Detroit earned a power play when Ryan Getzlaf was called for slashing at the 14:33 mark of the third period. Lidstrom then fired a slap shot that ricocheted off Holmstrom in front and caromed off the stick of an Anaheim defenseman into the net.
The game finished with Anaheim on a six-on-four advantage after pulling goaltender Jean-Sebastien Giguere.
Detroit will host Game 2 Sunday.