Vincent Kompany's first-half header secured the victory, which puts City ahead of their title rivals on goal difference, while the Red Devils were prevented from registering a single shot on target.
Hart told Sky Sports after the match: “I think everyone can feel [the intensity]. There's been a lot of build-up to this for weeks and weeks outside of the team.
“We've concentrated on what we've wanted to do – we've not been great at times but we're on a good run at the moment we've played a very good side there and beaten them.
“They're a good side and balls came flying in but when you've got the likes of Pablo Zabaleta, Kompany, Joleon Lescott and Gael Clichy – they were brilliant, so I always felt comfortable.”
But despite taking a narrow lead at the summit of the Premier League, Hart knows that his team still need to concentrate as they look towards a trip to face Newcastle on Sunday.
He added: “It doesn't change how we feel about this season. It won't change how we've gone into every game.
“It's great for us tonight but Newcastle is now our focus. They're really pumping themselves up in the league. St. James' Park is going to be tough and we're looking forward to taking a great support there like we've had great support at home and hopefully we can push on and get one game closer.”