Hamilton hit .177 in July, but had his second four-RBI game of the season. He has 10 RBIs in his past nine games.
"Basically I've said stop thinking about it, stop listening to everybody -- everybody's got something to say," Hamilton said. "Just go out there and play. React. Try to slow the game down again. It worked. There's nothing wrong with my swing, it's just about trusting myself, not being too aggressive and trying to slow the game down again."
The Rangers broke an 8-8 tie after six innings when they scored four runs in the seventh, including Hamilton's two-run double, and added three in the eighth.
Dempster allowed eight runs and nine hits in 4 2/3 innings. The Rangers acquired Dempster from the Cubs minutes before Tuesday's non-waiver trade deadline.
Roy Oswalt (4-2) threw two scoreless innings in relief for the win.
C.J. Wilson gave up eight runs and 10 hits in 5 1/3 innings. David Carpenter (1-2) took the loss for the Angels. Angels relievers were charged with 11 earned runs in 6 1/3 innings in the last two games of the series.
The Rangers earned a split of the four-game series with the Angels after Wednesday's night's 11-10 win in 10 innings.
"It was kind of a momentum kill from how we played the first two nights to how we played these last two," Angels manager Mike Scioscia said. "It's tough. It's a good team there and it's not a forgiving park if your guys don't hit their spots, and some of our guys in the pen had a tough time doing that."
It was the Rangers' first time scoring in double figures in consecutive games since May 7-8.
"It's been a lot of fun," Rangers second baseman Ian Kinsler said. "We're starting to show some excitement. We have a lot of momentum offensively right now. This was more of a prototypical Rangers game where we scored as many runs as we needed to win. We kept pounding them."
Texas leads Los Angeles by five games in the AL West.
Outfielder Torii Hunter led the Angels with three hits, two RBIs and a run scored. Kendrys Morales went 2 for 5 with a homer and two RBIs. Albert Pujols, Alberto Callaspo and Howie Kendrick added two hits each for Los Angeles.
Soto went 2 for 5 with two RBIs and two runs scored, and Olt, a rookie from the University of Connecticut, went 1 for 3 with a run scored.
Mike Olt was called up from Double-A Frisco Thursday and made his major league debut at first base just hours later. Rangers manager Ron Washington said Olt will mostly start at first base and also at designated hitter against left-handed starters. Olt can also play third base when Adrian Beltre has a day off. ... Angels outfielder Mike Trout was selected AL Player of the Month and AL Rookie of the Month for July. He's the first AL player to win both Player of the Month and Rookie of the Month in the same month. Trout also won the monthly rookie honor in May and June, joining Ichiro Suzuki (April, May and June 2001) as the only player to win in three straight months.