Byrd hit a single, a double and a solo homer and scored three times. Texas got a two-run double from Brad Wilkerson in the third, while three others drove in a run.
Ian Kinsler and Jarrod Saltalamacchia both had two hits for the Rangers, who have won four of five.
Winner Edinson Volquez (1-0) gave up three earned runs and seven hits in five innings.
C.J. Wilson recorded his ninth save after thwarting LA's ninth inning rally.
The Angels' Garret Anderson hit a sacrifice fly in the fifth and a three-run homer in the ninth. Howie Kendrick had three hits and two RBI and Maicer Izturis and Vladimir Guerrero both had two hits for the Angels, who had a five-game win streak broken.
The loser was Kelvim Escobar (15-7).