It was Detroit's fifth win in six games and it came with the help of 23 saves by Jimmy Howard, who held the Oilers scoreless through the first two periods.
The Red Wings scored the game's first five goals, with Cleary giving Detroit a 3-0 lead late in the first period and then building the advantage to 4-0 early in the second.
Holmstrom's first goal came with 6:31 to go in the opening stanza and he delivered the final Detroit score with 3:20 remaining in the third.
Howard did not allow a score until Dustin Penner came through on the power play 58 seconds into the third.
Edmonton has dropped four of its last five games. Nikolai Khabibulin surrendered all six goals on 34 shots. Khabibulin, however, did stop Mike Modano on a penalty shot late in the second.