City, who have been scoring for fun in the Premier League, needed a last-gasp goal from Sergio Aguero to secure a 2-1 win at home to Villarreal on Tuesday while United's nervy 2-0 victory at Otelul Galati came after two Wayne Rooney penalties.
Neither English side were convincing but the net outcome was the same - a first group victory allowing them to start thinking about qualifying for the knockout stage.
"It would have been very difficult [to qualify] without that goal," Mancini told a news conference where he surreptitiously glanced at the television screens showing replays of the stoppage-time winner and his manic touchline celebrations.
With City third in Group A on four points from three games, one behind second-placed Napoli and three behind leaders Bayern Munich, Mancini said topping the table was a realistic aim in his club's debut season in the competition.
His counterpart from the red half of Manchester, whose side are second in Group C which features teams who are on paper weaker than those in City's group, was also thinking about the next round with three games still to play.
"We won and that is the important thing," manager Sir Alex Ferguson said on the club's website.
"We have to do our job at Old Trafford, hopefully we can beat Galati again which brings us to eight points and I've always said that 10 points gets you through."
DIFFERENT STORY
City have raced to the top of the Premier League, where they have a two-point lead over their neighbours, by scoring on average more than three goals a game but it has been a different story on the European stage.
They blamed first-night nerves for a 1-1 draw at home to Napoli before slumping to a 2-0 defeat at Bayern where the furore over Carlos Tevez's alleged refusal to play diverted the spotlight away from a poor performance.
Mancini said his club were still getting used to juggling the Champions League with domestic competitions.
"It is not so hard for those clubs who do it every year but this is our first season and that is the hardest one of all," he said. "We need to keep improving and learning."
Three-times European champions United cannot hide behind the debutant excuse, having reached the Champions League final in three of the past four years, and for them it is not just their continental form that has been a problem.
Having set off at a blistering pace in the Premier League, scoring at an even higher rate than City, the English champions have stalled somewhat in the past month.
Giving away possession cheaply as they did in Romania on Tuesday has been a feature in the Premier League where they have also looked shaky in defence, an area of the team that has been hit by injuries.
Captain Nemanja Vidic's return from a calf injury should go some way to stabilising that but he is poised to miss at least the next Champions League game after being shown a red card against Galati.
