Texas committed four errors, three of them coming during a messy second inning during which the Angels scored three unearned runs.
Those miscues threatened to derail Texas' hopes of reaching the playoffs for a fourth consecutive year, but Profar came through with his game-winning shot off Michael Kohn (1-4).
With three games remaining in the season, Tampa Bay, Cleveland and Texas are battling for the two wild card positions in the American League. The Rays are out front of the Indians by one game and the Indians are one in front of the Rangers.
Tampa Bay and Cleveland have seven-game winning streaks working. Texas has won four in a row.
The Rangers took a 5-4 lead in the sixth on a two-run double by Leonys Martin, but Los Angeles tied it in the seventh on a triple from Kole Calhoun and an RBI single from Mark Trumbo.
The home run by Profar gave the win to Joe Nathan (6-2), who put a runner on third with one away in the top of the ninth before striking out Calhoun and Trumbo to end the threat.