The 21-year-old left-back appeared to be tripped in the box by Jordy Clasie early on in the second half.
Zlatan Ibrahimovic, who had opened the scoring before the break, converted the resultant spot-kick.
Shaw, a former Southampton player, was heckled by sections of the away support following the penalty, but later insisted that he had won it fairly.
"A lot of people are saying it wasn’t a penalty, but 100 per cent it was a penalty, he clipped my foot," he told MUTV.
"There was 100 per cent contact and I am not one to dive or to cheat. I am happy I played a part in one of the goals tonight."
Shaw, who was making his first appearance at Old Trafford since suffering a double leg-break eleven months ago, added that he believes United have started the season in title-winning form.
Jose Mourinho’s side have taken maxiumum points from their first two league games and lifted the Community Shield at Wembley in the traditional curtain-raiser.
"You can see tonight we can improve but we are winning games and we still have a lot to improve on," he said after Friday’s win.
"That is a sign of, you could say, champions," he added. “I don’t want to speak too soon, but we have got a very strong squad and hopefully we can keep training hard together, getting to know each other and push for the title this season.”
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies