The Reds dominated much of the game, however they failed to take the lead in the second half against Mourinho's side, who had equalised before the break after Sadio Mane's opener.
Jurgen Klopp withdrew Naby Keita for Shaqiri with 20 minutes to go as he looked for a winner that would see them go to the top of the Premier League.
The Swiss international took no time to make the breakthrough, netting just three minutes later to give the Reds a deserved lead.
He then added a third to seal victory for the Reds and speaking after the game, he was delighted with the impact he made for Klopp's side.
Speaking after the game, Shaqiri said: “Every player is not happy when he is on the bench, that's pretty normal. But only 11 players can play – the coach decides.
“I always try to make an impact when I come on or play from the beginning. Today was a very good impact with the two goals, I'm delighted.”
The Reds have now extended their unbeaten start of the season to 17 matches and see themselves at the top of the Premier League and Shaqiri insisted they must continue as they have done after Sunday's win.
He added: “It was an amazing win and important, too. We played very well. In the second half, we did more than Manchester United – we tried and tried. In the end, we fully deserved to win this game.
“When you see the statistics, we did everything in this game to win. It was very intense and we kept the high intensity in this game. It was an important win and a nice win against Manchester United.
“Our start to the season is amazing – a record in the club's history. But we have to keep going, we have to take it game by game.
“I hope we can have a lot of games like this and try to win a lot of games. If we play like this we can have an amazing season.”