Briceno, pinch-hitting for fellow catcher Francisco Arcia, hit a 0-2 pitch from Rangers pitcher Matt Moore (3-8) over the fence in left-center field to end the Angels' five-game losing streak.
Felix Pena started on the mound for the Angels and gave up three runs on five hits in seven innings. He issued one walk while striking out six. It was the third time in his past five starts that Pena went seven innings, his season high.
The Angels got two more solo homers off Rangers starter Adrian Sampson in the fourth, Marte's sixth of the season and Hermosillo's first in the major leagues.
Jurickson Profar's two-run homer in the top of the sixth tied the game at 3-3, but the Angels broke the tie in the bottom of the inning. Marte was hit by a pitch leading off, took second on a balk by Sampson and went to third on a groundout.
Sampson allowed four runs on five hits and two walks while striking out six in 5 1/3 innings.
Ty Buttrey took the ball for the Angels in the ninth inning with a chance to earn a save. However, with two outs, he gave up a double to Guzman and an RBI single by pinch hitter Elvis Andrus, tying the game at 4-4.