Rovers’ Norway winger Morten Gamst Pedersen ended Mancini’s perfect record since arriving with the second-half strike that pegged the Blues back to 3-1 before Tevez’s brilliant third.
City’s boss said: “It’s nice scoring goals but the most important thing was that we won - but we must have more attention in certain situations, because one lapse resulted in conceding a goal and the game changed after that.
“When I was in Italy I had six or seven Argentinian players, and they all told me that Carlos is a wonderful player, but I’d seen him enough times to know that for myself.
“Now I work with him on a daily basis and I can see how good he is in training and during matches. I hope he reaches any target he sets for himself, but of course I’d also like lots of other players to score goals too.”
The former Inter Milan manager, tasked with bringing Champions League football to the City of Manchester Stadium, added: “It is nice being in fourth now but it doesn’t count for anything at this moment. It’s important we finish in the top four at the end of the season.
“It’s important we continue our run and keep winning games if possible, and improve game after game. Every game is difficult and the next match at Everton will be tough too.
“We must prepare for every game very well otherwise it is difficult – preparation is everything. I want all my players to be playing well so I can bring players in and rest others when necessary.
“Vieira will be a good player for us but I didn’t want to risk him tonight because he needs a little more time to recover from his calf injury. He will do well for us, I’m sure.”