Magee, a native of Chicago, scored in the 23rd minute after Jovan Kirovski converted a midfield turnover into a counter-attack for the Galaxy (8-4-10).
The Galaxy was playing without Beckham after he earned a one-game suspension for a red card in Saturday's 2-0 loss to Seattle.
In the 54th minute, Magee sent a lead pass to Landon Donovan, who raced 50 yards past two defenders before scoring past goalkeeper Jon Busch.
Donovan started for the first time since being diagnosed with the H1N1 virus last week. The illness limited the Galaxy captain to 45 minutes off the bench Saturday.
The Galaxy moved past Seattle and into second in the Western Conference, five points behind first-place Houston.
The Galaxy lost goalkeeper Donovan Ricketts in the 85th minute after a collision with Fire forward Stefan Dimitrov. Ricketts was replaced by backup Josh Saunders.
The Fire, which failed to put a shot on goal, lost for the second time in three games. It remained second in the East, four points behind Columbus.
"We were all off," Busch told the Chicago Tribune. "Big players need to step up in big games, and none of us did. Effective soccer wins games and it's what L.A. just did to us. It's plain and simple."