The right-hander needed only 24 pitches to retire the first six batters he faced in Angel Stadium, including an 89-mph fastball that struck out Toronto slugger Jose Bautista to start the second inning.
But Weaver, who is 9-6 with a 3.50 earned-run average in 19 starts this season, tweaked his lower back while warming up before the third inning.
Manager Mike Scioscia and athletic trainer Adam Nevala came to the mound, and Weaver was pulled almost immediately in favor of right-hander Matt Shoemaker, who threw 102 pitches against the Houston Astros on Thursday night.
Scioscia said before the game that Shoemaker would start Thursday night at Texas, a decision that meant left-hander Hector Santiago would return to the bullpen.
But with Shoemaker thrust into an emergency appearance Monday night, Santiago, who is 0-7 with a 4.50 ERA, will likely start against the Rangers on Thursday night and remain in the rotation until Weaver returns.