The visitors should have taken the lead in the second minute of the game when a loose ball wasn't cleared and Landon Donovan's shot was well saved by Clinton Irwin.
The Rapids took advantage of their escape as they took the lead in the 20th minute through Sanchez, as he collected the ball on the right and chipped it over Jaime Penedo with a sublime effort.
Robbie Keane could have got the Galaxy level when he latched onto a cross into the penalty area, but he could only guide his head over the bar.
After half time, Los Angeles pressed forward and Irwin needed to be alert again to beat away a powerful effort from Dan Gargan after he had found space down the right.
The visitors were handed a golden chance to get back into the game when Moor brought down Rob Friend in the area and a penalty was awarded. Keane took the spot kick but he put his shot wide.
Deshorn Brown had the chance to wrap the game up for the Rapids with two late efforts but he couldn't beat Penedo, although it wasn't to matter as the home side held out for the win.