Beckham had played 90 minutes in England's 2-1 loss to Germany at Wembley on Wednesday, and arrived in Los Angeles with just hours to spare before starting Thursday's match.
In the wake of an 11-hour flight, Beckham took advantage of breaks in play to run to the touchline to swig water and sports drinks.
A minute before half-time, a brief brawl broke out after Chivas central defender Jesse Marsch caught Beckham with a waist-high tackle, and the Galaxy captain - whose entrance into the MLS has been hampered by an ankle injury - angrily confronted him.
As players from both teams rushed in, Galaxy midfielder Kevin Harmes threw a punch and was head-butted by Chivas defender Alex Zotinca. Both players were red carded by referee Terry Vaughn, and Marsch received a yellow card.
Beckham returned for the start of the second half, to the delight of the sell-out crowd at the 27,000-seat Home Depot Center, which is also Chivas' home ground.
The Galaxy fell behind in the 58th minute, when defender Kyle Veris and goalkeeper Joe Cannon failed to communicate on a long ball delivered from midfield and Cuban striker Maykel Galindo burst between the two and scored from the top of the penalty area.
In the 59th minute, Beckham was joined on the pitch by Landon Donovan, also returning from a Wednesday match in Europe - the USA's 1-0 loss in Sweden.
Still the Galaxy could make no headway, and Galindo scored a second goal in the 69th minute, breaking down the center of the field unmarked before slipping the ball underneath Cannon.
Francisco Mendoza made it 3-0 on a counterattack with two minutes left as Chivas took advantage of the Galaxy pushing up in an attempt to score.
Beckham was visibly tired even before the match was over, walking slowly with a slight limp. The result was a serious blow to the Galaxy, who remain in fifth place in the MLS Western Conference, 12 points out of the final play-off spot.