Sergio Aguero finally ignited City's European campaign by with an injury-time tap-in that secured a 2-1 win over the hard-working Spaniards at the Etihad Stadium.
It had looked like being another frustrating night for Mancini's men as they searched for their first Group A win after a draw and a defeat in their two openers.
Mancini said: "I hope this goal will be very important - it was very important to win the first game in the Champions League.
"I think everything can happen in this group, for first place.
"We know we should go to Villarreal in 15 days, it will be a very important game for us.
"Villarreal are a really good team and play football very well.
"In the Champions League when you get one chance you must score, otherwise the game can go the other way.
"We had a lot of chances and I think we deserved to win the game."
City fell behind in the fourth minute to a Cani strike and struggled to haul their way back into the game.
They eventually benefited from a stroke of luck when Carlos Marchena turned into his own net just before half-time and moments after Adam Johnson was surprisingly substituted for Gareth Barry.
Johnson was not injured and shook his head as he left the left but after the controversy over substitutions in City's loss to Bayern Munich last month, Mancini insisted there were no more problems.
He said: "We didn't have many players in the middle and I wanted to move Yaya (Toure) behind Edin (Dzeko).
"Adam played very well but at that moment I had to do a tactical substitution.
"I spoke with him (afterwards) and he understood this.
"Sometimes every manager decides to make a substitution. Unfortunately some player has to leave the pitch.
"I understand any player leaving the pitch after 38 minutes would be disappointed but Adam is an important player for us."
City's laboured performance was in contrast to their outstanding Barclays Premier League form but captain Vincent Kompany was in no doubt victory was justified.
The Belgian told Sky Sports: "It's totally deserved but unbelievable really.
"We're happy, maybe we're going to carry on doing these sort of things this season.
"I've said it before, I think we are in the toughest group. Villarreal have been in it [the competition] much more than us - so it makes it a beautiful night."
Mancini was reluctant to discuss the outcome of tomorrow's investigation into Carlos Tevez's alleged refusal to play as a substitute in last month's Champions League match at Bayern Munich.
However, he intriguingly added: "I'm soft because I'm a good man and I can't understand everything but I would prefer not to speak about this situation."
Villarreal coach Juan Carlos Garrido admitted defeat was hard to take.
Garrido said: "I think we have done a good match and we did deserve - if not to win - a 1-1 (draw).
"I thought we had a good match in general, in all aspects, and to lose in the last minute in this way is difficult.
"Congratulations to City because they won but we thought we deserved more from this match.
"My players thought the goal was offside but it was not offside. They thought it was unfair to lose this way but we must accept it."