Dempster (2-1) yielded one run, walked two and struck out six after returning to the team following a one-start absence for a personal leave.
The right-hander, obtained from the Chicago Cubs in a trade-deadline deal, had given up eight runs in two of his three previous Texas starts but handled the Orioles Monday, tossing 66 of his 111 pitches for strikes.
Miguel Gonzalez (5-3) was touched for four runs on eight hits over five frames for the Orioles, who came into the contest holding the second American League wild-card spot.
Adam Jones supplied Baltimore's only RBI with a first-inning single that drove in Nate McLouth.
UPI horse racing weekend preview
Bo Jackson: I can't help Jameis Winston anymore