Dempster (2-1 in Texas, 7-6 overall) retired the last 11 batters he faced. The right-hander acquired from the Chicago Cubs on July 31 allowed one run and four hits.
David Murphy had two RBI singles for the Rangers, who opened a 10-game homestand. All nine Texas starters had a hit.
Miguel Gonzalez (5-3), who had won his previous three decisions, gave up four runs in five innings.
The Orioles have still won 11 of their last 16 games.
In his 16 starts for the Cubs before the trade, Dempster had a 2.25 ERA and pitched eight innings twice. After giving up eight runs twice, Dempster had an 8.31 ERA in his first three starts for Texas.
RAYS 5, ROYALS 1
ST. PETERSBURG, Fla. (AP) — Tampa Bay's Jeremy Hellickson pitched seven innings for his first home win in three months.
Hellickson (8-8) allowed one run and six hits in ending a stretch of seven consecutive home winless starts, dating to a 2-1 victory over Boston on May 16.
Tampa Bay was coming off an 8-2 road trip that concluded with a four-game sweep over the Los Angeles Angels.
The Rays went ahead 4-1 in the fourth on Jose Labaton's RBI single and a run-scoring double from Desmond Jennings off Will Smith (4-5).
Tampa Bay reliever J.P. Howell worked a scoreless eighth to extend his team record streak to 25 2-3 innings. Kyle Farnsworth got the final three outs in a non-save situation.
Smith lasted 3 2-3 innings, giving up four runs and eight hits.
WHITE SOX 9, YANKEES 6
CHICAGO (AP) — Alexei Ramirez hit a tiebreaking two-run homer in the seventh, Adam Dunn added his major league-leading 36th an inning later and Chicago ended a three-game losing streak.
Ramirez connected with two outs in the seventh off Boone Logan (4-2) for his seventh homer of the season. Brett Myers (2-1) got the win by pitching two-thirds of an inning. Addison Reed worked the ninth for his 22nd save in 25 chances as the White Sox won the opener in a three-game series between two first-place teams.
Dewayne Wise and Gordon Beckham also homered for Chicago.
New York's Derek Jeter homered and had four hits, tying Eddie Murray for 11th place on the career list (3,255). He also took over 13th place on the runs scored list (1,845), passing Craig Biggio.
Wise hit a two-run homer for Chicago as the White Sox scored five times in the fifth for a 5-3 lead. The Yankees went ahead in the top of the sixth on Jeter's solo homer and RBI singles by Mark Teixeira and pinch-hitter Casey McGehee.
MARINERS 5, INDIANS 3
SEATTLE (AP) — Michael Saunders hit two two-run home runs to help Seattle to its sixth straight victory.
Saunders hit a two-run homer Sunday and has a career-high 13 homers.
Eric Thames had a solo shot for the Mariners in the sixth inning as well as a triple in the eighth.
The Mariners have won 13 of their last 14 games at Safeco Field. It's the best home stretch over 14 games in club history. They are 23-13 since the All-Star Break.
The Indians have lost six straight.
Charlie Furbush (5-2) earned the victory in relief of starter Kevin Millwood.
Tom Wilhelmsen worked the ninth and had trouble closing. Casey Kotchman singled with one out and Ezequiel Carrera doubled to right, with Kotchman taking third. Pinch-hitter Jack Hannahan loaded the bases with a walk.
Jason Kipnis, who had two RBIs, then grounded into a double play. Wilhelmsen picked up his 18th save in 20 opportunities.
Joe Smith (7-4) took the loss in relief of starter Ubaldo Jimenez.
TWINS 7, ATHLETICS 2
OAKLAND, Calif. (AP) — Josh Willingham drove in two runs against his former team and Joe Mauer hit a two-run single as Minnesota snapped Oakland's four-game winning streak.
Brian Duensing (3-8) struck out seven and walked one in six innings to help the Twins bounce back from a three-game weekend sweep at Seattle that began a 10-game road trip.
Darin Mastroianni had an RBI single and a sacrifice fly and No. 9 hitter Pedro Florimon joined Willingham and Mauer with three hits apiece for Minnesota, which went 4 for 11 with runners in scoring position a day after missing in all 10 such chances at Seattle.
Oakland had its five-game home winning streak snapped with just the club's 11th defeat in 37 games at the Coliseum since June 6.
A's starter Brandon McCarthy (6-5) surrendered three straight one-out hits in the first, including Willingham's RBI double. Trevor Plouffe drew a bases-loaded walk as the Twins took a quick 2-0 lead.
McCarthy was knocked out after 3 1-3 innings. He allowed 10 hits and a season-high six runs.