Sabah scored in injury time after Adrian Aldrete equalized in the 83rd.
Former Tottenham player Robbie Keane gave the Galaxy the lead in the 52nd minute. Morelia had 70 percent of the possession in the first half, but it was the Galaxy that finally broke through with the goal.
Galaxy's Omar Gonzalez had the best chance in the first half with a 31st-minute header that went off the crossbar. Keane got Galaxy on the board, taking a pass from Landon Donovan and slotting the ball home.
Aldrete took advantage of a goalkeeping error to level the score when his long shot slipped through the hands of Galaxy keeper Josh Saunders. Sabah then sealed the win with a header inside the near post.
Morelia had 70 percent of the possession in the first half, but former Tottenham player Robbie Keane gave the Galaxy the lead in the 52nd minute.
Morelia now shares the Group A lead with the Galaxy on six points from three matches.