Brodeur returned to the lineup on Thursday night. He had been out since Feb. 21 with a pinched nerve in his back and neck.
With a delayed penalty call pending against New Jersey's Marek Zidlicky, Staal passed the puck back to the point from behind the net.
The pass missed intended target Tim Gleason and bounced off the boards near center ice. Carolina goalie Dan Ellis had already skated toward the bench because of the delayed penalty and couldn't get back in time to prevent the puck from sliding into the net.
Because he was the last Devils player to touch the puck, Brodeur got credit for a power-play goal.
Brodeur is the first NHL goalie credited with three goals, according to ESPN Stats and Information.
The Devils won the game 4-1.
Information from the Associated Press was used in this report.