Jason Vargas limited the Dodgers to five hits over seven innings to get the win, which was the Angels 10th in 12 games.
The Dodgers tied it at 2-2 in the fifth on a home run by Ramon Hernandez, but the Angels moved back in front an inning later.
Howie Kendrick doubled with one out in the sixth and, with two away, stole third. Kendrick came dashing home on Nelson's hit, one of nine the Angels managed against three Dodgers pitchers.
Vargas (5-3) struck out six while throwing 97 pitches. Ernesto Frieri gave up a leadoff single in the ninth, but then fanned two to collect his 12th save.
Ted Lilly (0-2) was tagged with the loss, giving up three runs on five hits in 5 2/3.
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