Playing in front of a sellout crowd of 27,000 at the Home Depot Center, MLS-leading Los Angeles improved to 6-2-5 and ran its unbeaten streak against its area rival to nine games - four victories and five ties.
Beckham has six assists this season to match Houston's Brad Davis for the MLS lead.
''The game showed the character of the team to dig in when we did, we knew they were going to come forward and try to attack us in many different years, but to get the 1-0 win is important,'' Beckham said.
Beckham was a force throughout the night, but will miss the club's next game against Houston to attend a testimonial match for former Manchester United teammate Gary Neville at Old Trafford in Manchester, England, on Tuesday, Galaxy coach Bruce Arena announced following the match.
''The issues are outside of this team because there are no issues inside of it,'' Beckham said. ''If the fans have a problem with it, then they can ask Bruce Arena, the manager, who doesn't have a problem with it or any player in this locker room, who don't have a problem with it.''
Chivas USA (3-4-3) had its best scoring chance in 65th minute when midfielder Ben Zemanski fired a shot that bounced off the post. Chivas USA hasn't beaten the Galaxy since Sept 13, 2007.
''We didn't play well enough for 90 minutes, we didn't play our game for the first half and that hurts you,'' Zemanski said. ''We weren't good with the ball and defensively we didn't do well to keep them behind us so it really hurt us.''