Los Angeles fell behind 1-0 in the 53rd minute on a Joao Plata goal, but then Keane erupted, scoring in the 56th and 67th to hoist the Galaxy into the lead.
Omar Gonzalez made it 3-1 for the Galaxy in the 75th minute and Keane struck again in the 86th minute, giving him 10 goals for the season.
Los Angeles (11-9-4) used the convincing win to pull within four points of Western Conference-leading Salt Lake.
Robbie Findley also scored for RSL (12-8-5), whose lead over second-place Colorado withered to two points.
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