Keane found the net on opposite sides of halftime, notching his fifth and sixth tallies of the season as defending MLS champion Los Angeles (7-10-2) got back into the playoff picture.
The victory allowed the Galaxy to climb two spots into fifth place in the Western Conference, totaling 23 points.
Chicago, which holds fourth place in the Eastern Conference, fell to 8-6-4.
Millions of Getty images now available for free via embed tool
Aaron Carter is still in love with Hilary Duff