The Angels (72-49) are alone in first place in the AL West for the first time since 2011, but they are only two percentage points ahead of the Oakland Athletics, who lost at Atlanta on Saturday.
Los Angeles has won nine in a row over Texas, tying the series record set in 1985 and matched a year later.
Huston Street worked a scoreless ninth inning for his 33rd save in 34 opportunities. It was the second straight night the Rangers rallied before losing 5-4.
Street hasn't allowed a run in 12 games with the Angels since being acquired from San Diego on July 18.
Pujols hit a solo homer in the first against starter Colby Lewis (8-10). Kendrick hit another solo shot in the second.
Los Angeles' Chris Iannetta doubled home another run in the second, and Calhoun singled to drive in two more in the fourth. Both hits, like Pujols' homer, were with two out.
The Rangers pulled within 5-2 in the fourth on J.P. Arencibia's run-scoring single that broke an 0-for-19 slump. Their final runs came in the seventh when Alex Rios snapped an 0-for-16 drought with a two-run single.
Lewis allowed five runs and 10 hits in six innings. He struck out 10 for the first time since June 17, 2012.
Angels: Los Angeles activated RHP Mike Morin (left foot laceration) from the disabled list. He pitched an inning in relief.
Rangers: Rios made his fourth consecutive start after aggravating a sprained right ankle on Aug. 10. Texas manager Ron Washington said Rios would be the DH on Sunday. . RHP Yu Darvish (inflamed elbow) won't travel with the team to Miami on Monday, and will continue treatment in Arlington.
The teams' three-game series concludes on Sunday. Angels LHP Hector Santiago (3-7, 5.84) defeated the Rangers 15-6 on July 10 in Arlington. He pitched six scoreless innings and received a no-decision Aug. 10 vs. Boston.
Texas RHP Nick Tepesch (4-7, 4.29) last faced the Angels on July 11, when he lost 3-0. His first career start against Los Angeles was a no-decision.