Hamilton, who insisted beforehand he had no extra emotions facing the Angels, went 1 for 3 with a double and two strikeouts.
Erick Aybar had his first career five-hit game and scored twice. His two-run single in the second, his second hit that inning, made it 6-0 before Hamilton had even gotten an at-bat for Texas.
Garrett Richards (9-5) won his sixth consecutive start against the Rangers. The right-hander struck out six and allowed one run over 7 2/3 innings.
Rangers rookie right-hander Chi Chi Gonzalez (2-4) gave up six runs in 1 2/3 innings.
The Angels matched a season high with 15 hits and won for the fifth time in six games, a stretch when they have overtaken Texas for second place in the AL West.
Calhoun's three-run double in the second ended an Angels slide of 25 hitless at-bats with runners in scoring position. He added an RBI double in the third on a play when leadoff hitter Johnny Giavotella was also thrown out trying to score from first.
Hamilton played only his 10th game this season for Texas, his third since being activated this week after a month on the disabled list with a left hamstring strain. He is hitting .303 (10 for 33) with two home runs.
Ryan Rua, who replaced Hamilton in the field to start the eighth, homered leading off the ninth.
Los Angeles traded Hamilton on April 27, after an offseason that included shoulder surgery and a self-reported substance abuse relapse. The 2010 AL MVP played only two seasons with the Angels after signing a $125 million, five-year deal during free agency after the 2012 season.
Texas is responsible for $6 million through 2017 after Hamilton agreed to give up $14 million he was due from the original deal.
Hamilton said before the game he has moved forward since returning to the franchise he led to a pair of World Series appearances.
"There's no searching or closure for me. It's over. I've moved on. I'm in Texas now," he said. "I've got a great group of guys here. I had a great group of guys in Anaheim."
Angels: RHP Jered Weaver (inflamed left hip) played catch on the field before the game. He is eligible to come off the DL on Monday, but manager Mike Scioscia said he will need a rehab start and likely won't pitch for the Angels until after the All-Star break. ... RHP Cory Rasmus (left abdominal surgery) came off the 60-day DL and was optioned to Triple-A Salt Lake City, where he has been on injury rehab since June 19.
Rangers: Manager Jeff Banister said OF Delino DeShields (left hamstring strain) has looked good in his rehab stint at Triple-A Round Rock. Banister said there is a chance DeShields could be back before the All-Star break.
Angels: Left-hander Hector Santiago (4-4) starts against Texas for the third time this season. He allowed only two earned runs in 12 1/3 innings in the first two games.
Rangers: Wandy Rodriguez (5-3) has won four of his last five decisions, but gave up a season-high eight runs in his last home start June 24 against Oakland.