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.
Benedict Cumberbatch's dramatic reading of R. Kelly lyrics is just what you need
Gal Gadot cast as Wonder Woman for 'Batman vs. Superman'