It was the fifth start for Greinke since he was acquired by the Angels. In his first four, he had turned in a 7.20 earned run average.
He allowed five hits and struck out five while helping the Angels win their fourth in a row.
Downs retired the last batter of the eighth and the first of the ninth before Garrett Richards came on to strike out the only two men he faced for his first major league save.
Howard Kendrick snapped a scoreless tie in the sixth with a two-run double.
The Tigers had a three-game winning streak halted.