Bourjos entered the game after Kendrys Morales led off the ninth inning of a 5-5 game with a double off Josh Kinney (0-2).
Then with Maicer Izturis at the plate, Kinney sailed a ball wide of the plate that went off catcher John Jaso's glove. Jaso retrieved the ball and fired to Kinney covering the plate, but Bourjos beat the throw for the game-winner.
Ernesto Frieri (1-0) collected the win after striking out the side in the top of the ninth.
Jaso smashed a three-run homer and Dustin Ackley added a two-run shot for the Mariners, who dropped the first of a three-game series.