A Vincent Kompany header just before halftime was enough to help City beat United 1-0 at the Etihad Stadium on Monday evening and, in the process, move above United at the top of the table on goal difference.
However, despite admitting his side deserved the win, Mancini has warned his players to be on their mettle against the Magpies on Sunday, with United facing an "easy" home fixture against Swansea.
"We are happy but I think that next Sunday we will have another difficult game," he told Sky Sports 1.
"They [United] play an easy game against Swansea. I think at this moment we are top of the league, but United has a slight advantage yet."
He added: "We deserved to win this game. We scored, we played well. We had other chances to score a second goal.
"I don't think they had a chance to score. But it was a difficult game because United is a top team."
After scoring what proved to be the winner, City captain Kompany said City's players are now on the brink of realising their childhood dreams, but called for focus for the remainder of the season.
"Absolutely buzzing. We've been waiting for this moment," he revealed.
"It's far from over and we know that, but to give our fans two wins over United this season... we have to finish it off and we have to do it over the next two games."
The euphoric City skipper added on Sky Sports 1: "Someone texted me today [Monday] and said he thought I was going to score the goal, and I thought he was a lunatic!
"For us to get this result, we wanted it so bad.
"It all comes down to the dream we had when we were children, and we all want it so much."