The loss by the Angels, who failed to score after loading the bases in the eighth, secured Texas' second straight AL West crown. They also fell 3 1/2 games behind the wild card-leading Boston Red Sox with five games to play.
Gio Gonzalez (15-12) outdueled Weaver (18-8), allowing one run and three hits over 7 1/3 innings, and received immediate support from Weeks, who led off the game by sending Weaver's third pitch over the right-field wall.
"Jemile's leadoff homer set the tone right off the bat," Gonzalez said. "I did my best to keep us in the game. I was extremely focused and wanted to pound the strike zone."
The Angels tied the game in the seventh when Torii Hunter smashed his 22nd home run of the season to right-center.
But Oakland quickly retook the lead in its next at-bat. Kurt Suzuki singled leading off the eighth and Scott Sizemore bunted him into scoring position. A wild pitch by Weaver moved Suzuki to third and he scored the go-ahead run when Weeks' grounder couldn't be fielded cleanly by Los Angeles third baseman Maicer Izturis.
DeJesus provided insurance by going deep with one gone in the ninth, the final batter Weaver faced.
"Jered pitched well and gave us a chance to win," said Angels manager Mike Scioscia.
Weaver gave up six hits and fanned eight without issuing a walk.
Oakland is 10-7 against the Angels this year...Both teams were 0-for-4 with runners in scoring position...Andrew Bailey picked up his 22nd save.