Beavan (10-10) scattered all eight Texas hits, issued one walk and struck out two en route to the win, tossing 65 of his 93 pitches for strikes.
Josh Kinney threw a scoreless eighth and Tom Wilhelmsen earned his 29th save for the Mariners, who clinched their three-game series.
Michael Saunders provided the game's only run with a second-inning homer.
Matt Harrison (17-10) absorbed the complete-game loss for National League West-leading Texas, allowing one run on seven hits. The left-hander walked four and struck out six.