The four-run sixth helped Toronto erase a 3-2 deficit.
Winner A.J. Burnett (13-9) allowed four earned runs and six hits in 5 2/3 innings.
Closer B.J. Ryan earned his 21st save with a scoreless ninth.
Frank Catalanotto hit a two-run double in the fifth for Texas. Gerald Laird had two hits and one RBI and a run scored on Barajas' second-inning throwing error.
Three players had two hits in the Rangers' second loss in their last three games.
The losing pitcher was starter Scott Feldman (4-4).