The Brazilian midfielder, who has featured in United's last eight games including the victory at the Madejski Stadium on Saturday, was forced off just before half-time having netted United's first goal of the game with an excellent drive into top corner.
Sir Alex told MUTV after the match that Anderson had pulled his hamstring and now faces a spell on the sidelines.
"It's a hamstring unfortunately. He'll be out for a few weeks," Sir Alex confirmed.
"He's been playing really well and scored a great goal today. I'm very disappointed for him."