Keane equalised for the Galaxy in the 56th minute when he received a long ball from Juninho whilst getting rid of his marker Carlos Salcedo before cutting back and finishing neatly from the edge of the area.
The former Tottenham man then edged LA ahead in the 67th minute when he slotted home a front post volley from Sean Franklin's delivery before making it 4-1 with four minutes remaining, after Omar Gonzalez's header, with a deft chip over RSL keeper Nick Rimando.
The trio of goals took Keane's season tally to 10 from 14 outings and was his second hat-trick of the campaign following on from his treble against the Seattle Sounders in May.
2013 MLS Player of the Week
Week 25: Robbie Keane (LA Galaxy) Week 24: Landon Donovan (LA Galaxy) Week 23: Lloyd Sam (New York Red Bulls) Week 22: Álvaro Saborío (Real Salt Lake) Week 21: Chris Rolfe (Chicago Fire) Week 20: Camilo (Vancouver Whitecaps) Week 19: Brad Knighton (Vancouver Whitecaps) Week 18: Alan Gordon (San Jose Earthquakes) Week 17: Camilo (Vancouver Whitecaps) Week 16: Kenny Miller (Vancouver Whitecaps) Week 15: Olmes García (Real Salt Lake) Week 14: Diego Fagundez (New England Revolution) Week 13: Marco Di Vaio (Montreal Impact) Week 12: Eddie Johnson (Seattle Sounders) Week 11: Will Bruin (Houston Dynamo) Week 10: Graham Zusi (Sporting Kansas City) Week 9: Tim Cahill (New York Red Bulls) Week 8: Jack McInerney (Philadelphia Union) Week 7: Brad Davis (Houston Dynamo) Week 6: Ryan Johnson (Portland Timbers) Week 5: Will Johnson (Portland Timbers) Week 4: Juan Agudelo (Chivas USA) Week 3: Bill Hamid (D.C. United) Week 2: Robert Earnshaw (Toronto FC) Week 1: Mike Magee (LA Galaxy)