Ruud Gullit's first MLS game in charge of LA Galaxy ended in a 4-0 humiliation at Colorado Rapids.
Cooke, 31, made his MLS debut in 2005 following two years with Sheffield Wednesday, having emerged through the United youth system but failing to make the grade at Old Trafford.
Beckham, who played in just five games last season with the Galaxy after joining the club in July, had a strong first half, delivering a series of deft passes, but his team stalled badly after half-time.
The former United and Real Madrid player won his 100th England cap against France in midweek and insists he would like to continue his international career.
"We've got a good team, we've got good players, we just need to take the positive from this and move on," Beckham said afterwards.