Beckham sent in a cross from the sideline in the 58th minute that Donovan headed into the net from the back post to make it 1-0.
Cristian Nazarit scored in the 62nd minute for Chicago (2-5-12), using his head to finish a cross from Patrick Nyarko.
Beckham's kick from the left corner in the 66th minute that skipped untouched through a pool of players to help the Galaxy (10-2-9) extend their league-best unbeaten streak to 6-0-6.
Tempers flared in stoppage time when the Galaxy's Chris Birchall made a hard sliding tackle on Nyarko. Players from both sides came together near the sidelines before order was eventually restored.