Gibson, 19, opened the scoring twenty minutes from time with a sizzling left-foot drive which gave Rovers' substitute stopper Ben Smith no chance.
He had earlier struck the bar with an equally impressive hit and Sir Alex is hoping to see the young Irishman build on his solid performance at the Keepmoat Stadium.
"I thought Darron was excellent," he told MUTV. "He is potentially one of the best young players at the club. He's got good physical prowess, two great feet, good pace - he's got everything in his locker.
"He just needs experience and that's something he'll hopefully get plenty of this season."
The United boss was pleased with his young team's overall display and believes debutant Anderson will have picked up some useful experience during his first half run-out.
"It was a good introduction to English football for him because Doncaster were very competitive and showed great commitment and made things difficult for us," explained the boss.
"He's picked up a little hamstring strain which we'll assess on Saturday. But hopefully it's nothing serious and he'll be okay."