Martinez, making his debut in a Galaxy uniform, opened the scoring in the 44th minute when he took a pass from Cobi Jones, drove past defender Chad McCarty and beat goalkeeper Tom Presthus.
He was acquired in a five-player trade from Dallas on July 11.
Lalas extended the lead in the 52nd minute, heading in a shot off a free kick by Simon Elliott.
Los Angeles (4-6-7) won for just the second time in its last seven games.
Brian Maisonneuve tallied in the 79th minute for the Crew (5-7-5), which had its winless streak extended to four games.