The striker, who completed a £16m switch to the Gunners on transfer deadline day, scored twice as England beat Switzerland 2-0 in their opening Euro 2016 qualifier last night.
"Man United will regret selling Danny Welbeck to Arsenal," Butcher told talkSPORT. "He's got blistering pace, is a good team player and finishes very well.
"The way Arsenal play, and the chances they'll give him, will make Man United regret their decision. He's enjoying his football at the moment. Arsene Wenger must be purring because it is a superb piece of business."
Welbeck has now scored 10 goals in 28 appearances for the Three Lions.