The Angels won their sixth in row and own the best record in the majors at 83-53. They shut out the A's for the 29 consecutive innings while opening a five-game lead atop the AL West heading into September.
Trout had a two-run single during a six-run second inning. Chris Iannetta homered and had three hits for Los Angeles, which has won 15 of 19. The A's have lost 14 of 20.
Scott Kazmir (14-7) got just four outs for Oakland, which scored its first run since Thursday on Josh Reddick's RBI single off Mike Morin in the eighth.
The sellout crowd loudly chanted "Sweep! Sweep!" as the club finished off its most impressive weekend of a remarkable season.
The weekend was awful for the A's, who got slugger Adam Dunn from the Chicago White Sox in a trade Sunday to help their slumping offense. Oakland, which led the AL West for nearly four straight months this season, now faces its biggest division deficit since May 25, 2013.
After eight Angels relievers pitched a three-hit shutout Saturday night, Shoemaker (14-4) provided the bullpen with plenty of rest.
The rookie right-hander capped a tremendous month with his sixth victory of August, striking out seven and extending his personal scoreless innings streak to 23 1-3 innings while allowing no Oakland runners to reach third base.
Gordon Beckham and Kole Calhoun drew bases-loaded walks in the second from Kazmir, who matched the shortest start of the season. A's manager Bob Melvin was ejected for apparently arguing balls and strikes after pulling Kazmir.
Trout led off the seventh with a home run, tying his career high with his 97th RBI. Iannetta added a homer in the eighth as the Angels outscored Oakland 18-4 in the series.
Erick Aybar extended his hitting streak to a career-best 16 games with a second-inning single to shallow left, and Josh Hamilton ran straight through third base coach Gary DiSarcina's stop sign to score Los Angeles' first run.
Kazmir then issued three straight walks and got yanked by Melvin, who was tossed after an animated conversation with home plate umpire Gerry Davis.
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