City prevailed 1-0 thanks to captain Vincent Kompany's first-half header to take them top of the Premier League table.
The managers had to be kept apart on the touchline by fourth official Mike Jones after a difference of opinion over Nigel De Jong's foul on Danny Welbeck led to angry words and gestures being exchanged.
Ferguson said: "He was badgering the referee the whole game, the fourth official and the linesman, and the minute I go out there... He was complaining about referees all week, he can't be complaining tonight, that's for sure."
Victorious Mancini played the incident down, adding: "I don't remember (what the row was about). I told the fourth official it was a foul for us, not against.
"At that moment the tension was high, but for me it's forgotten."
City, eight points adrift after the Easter fixtures having been in charge for so long, seemed to have had their title dreams shattered but they have dragged themselves back to the summit on goal difference with two games remaining and are within sight of a first title since 1968.
United, bidding for a record 20th English crown, have stumbled, losing at Wigan Athletic and then throwing away a 4-2 lead against Everton last weekend.
They still boasted a three-point lead before kick-off on Monday but were strangely subdued and barely tested City's England keeper Joe Hart as the team Ferguson once labelled the "noisy neighbours" seized the title initiative with a priceless 1-0 victory to seal a league double over their arch rivals.
Mancini expressed some surprise at United's cautious approach although admitting the job was far from complete.
"It's a great victory but we have two difficult games left and United have two easy games," said Mancini, whose side face fifth-placed Newcastle United this weekend before a final home game against strugglers Queens Park Rangers.
"United are still favourites. (United's opponents) Swansea and Sunderland have nothing to play for. I am very happy that we are top but it doesn't change anything.
"United put all their players behind the ball and only one striker. They wanted the draw. They played a different game to usual. We deserved to win."
Ferguson, who began with Wayne Rooney as a lone striker, said his side underperformed and the title was now in City's hands.
"It was feisty and competitive as we expected and there were not a lot of goal chances," he said. "David de Gea only had one save to make but I'm disappointed we never tested their goalkeeper to be honest.
"It was a damaging result, they are in the driving seat and we are up against it. They only need to win two more games and they win the league, simple as that.
"It's not over yet, of course it's not over, there is still football to play. But the situation is that they have an eight goal advantage which is a big advantage at this stage."
After a bright start United played second fiddle as City upped the tempo and began to dominate possession with former United striker Carlos Tevez driving his side forward.
United would have been happy to reach half-time at 0-0 but City captain Vincent Kompany produced a thumping header a minute before the break to put his side in control.
"If you let in a goal from a set piece at this level of football you only have yourselves to blame," Ferguson said.
"It was a bad time to concede because there was nothing happening in the game. They had lots of the ball around the box without doing anything then they get a corner kick and score."