Franklin scored a spectacular goal in the 49th minute, firing in a laser shot just inside the post off a feed from Robbie Keane.
The tally gave the two-time defending MLS champions a one-goal advantage heading into Thursday's second leg of the total-goal Western Conference series at Salt Lake City.
Nick Rimando made one save for RSL.
Martin Bashir resigns from MSNBC over Sarah Palin comments
Florida bear attack: Black bear mauls woman's face