Weaver (7-3) was extra tough, yielding only two hits, walking none striking out 11 en route to winning a second consecutive start for the first time this season.
Scott Shields finished the Angels' shutout with two more scoreless frames.
Howie Kendrick homered to lead off the game, keying a four-run opening-inning rally for Los Angeles highlighted by a bases-clearing, three-run double by Juan Rivera.
Torii Hunter added a two-run shot to lift the Angels to their second straight inter-league win over Chicago.
Cubs starter Ted Lilly (2-6) was shelled after taking a no-hitter into the ninth inning in his previous start. He surrendered eight runs on nine hits over 5 1/3 innings.
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