The Astros became the first team in the American League to lose 50 games.
Houston's first run came on a homer by Jason Castro in the fourth and the Astros held the lead until the sixth, when Mike Trout's RBI single tied it.
Los Angeles then took the lead in the eighth with Hank Conger's leadoff double leading to the tiebreaking run.
Bourjos came on to run for Conger and advanced to third on a fly ball to J.B. Shuck. When a relay throw to third by shortstop Ronny Cedeno sailed wide for an error, Bourjos dashed home to make it 2-1.
Eric Aybar's two-run single in the ninth gave the Angels some insurance, which became vital when Carlos Pena scored for the Astros on a sacrifice fly in the bottom half of the inning.
Dane De La Rosa (3-1) worked two-thirds of an inning in relief of Jerome Williams to get the victory. Ernesto Frieri survived a shaky ninth for his 20th save.
Reliever Paul Clemens (4-3) gave up the go-ahead run in the eighth to take the loss.