Kinsler's sharp single to right was fielded by Hamilton, who short-hopped a throw to the plate as speedy Craig Gentry slid home for the tiebreaking run.
Tanner Scheppers (1-0) pitched an inning in relief of starter Derek Holland. Joe Nathan worked the ninth for his first save, an inning that started with him retiring Albert Pujols and Hamilton on flyballs.
Gentry reached with a one-out single in the eighth off Scott Downs (0-2), the fourth Angels pitcher, and stole second base. After pinch-hitter Jeff Baker struck out, Kinsler put the Rangers ahead.
It was the fifth home opener in a row in which Kinsler got a hit and drove in a run.
Hamilton, now 1 for 16 this season, was booed during pregame introductions and when he came to bat each time. Those boos became cheers when took a strike on the first pitch in the second inning, and the sellout crowd of 48,845 erupted when he struck out swinging.
After striking out again in the fourth, Hamilton lined out in the sixth and flied out in the ninth.
Chris Iannetta homered and scored both runs for the Angels, who have lost three in a row since a season-opening victory at Cincinnati Hamilton's other former major league team.
Adrian Beltre homered in the seventh for Texas, which has won three in a row since dropping the season opener at Houston. Nelson Cruz and A.J. Pierzynski each had three hits.
Holland, who pitched for Team USA in the World Baseball Classic, allowed two runs over seven innings. The left-hander struck out five and walked one.
Jason Vargas allowed one run over 5 2-3 innings in his first start for the Angels, who acquired the left-hander last winter when they traded Kendrys Morales to Seattle. Vargas scattered eight hits with four strikeouts and two walks.
The Rangers were playing their first game at home since a 5-1 loss to Baltimore last October in the one-and-done AL wild-card game. Hamilton was also 0 for 4 with two strikeouts in that game when he was booed by the home fans who had cheered him through five All-Star seasons and the franchise's only two World Series appearances.
Iannetta's second homer of the season tied the game at 1 in the third.
Alberto Callaspo and Iannetta had consecutive singles to start the fifth. Callaspo was picked off second base by Holland, then thrown out trying to advance while Iannetta went to second. Mike Trout doubled off the left-field wall for a 2-1 Angels lead.
Cruz has had multiple hits in every game after getting two hits in each of three games at Houston.
Cruz had a leadoff double in the second and scored when Pierzynski, the catcher playing his first home game for the Rangers, tripled into the right-field corner. The ball ricocheted off the wall past Hamilton, and Pierzynski slid into third base at the same time the tag was being applied.