Just as in the weekend's Rome derby in Serie A, Mancini headed home the winner from a Paulo Dybala corner to earn Roma a precious victory against their fellow Italians.
Mancini has had an eventful few days as he was fined 5,000 euros ($5,427) on Tuesday for waving a flag depicting Lazio supporters as rats in his post-derby celebrations.
The 27-year-old's sixth goal of the season, in the 17th minute, gave his team a hard-fought victory and a lead to defend at next week's second leg in Rome, when Roma will continue their bid for a third European final in as many years.
"I'm so happy. I've been spoken about a lot in the last few days but to win the derby and come here and win in the way we did, like a proper team, is a huge joy," said Mancini to public broadcaster 'Rai'. "It's a great night for us. I couldn't have asked for more from this week."
Thursday's win also continued former Italy midfielder Daniele De Rossi's remarkable start as coach of his boyhood club who beat Milan at the San Siro for the first time since October 2017.
De Rossi has completely revitalised Roma since taking over from Jose Mourinho, who was dismissed in January after a league thumping at the hands of Milan.
Roma have only lost twice under his charge, to Italian champions-elect Inter Milan and Brighton in the previous round of the Europa League when that last-16 tie was decided in the first leg.
Milan, who struck the woodwork late on through Oliver Giroud, now face an uphill task to get something from their season as they trail Inter by 14 points in Serie A.
Stefano Pioli's side, who came into Thursday's clash on a run of seven straight wins, could even concede the league title to their local rivals in the next derby with Inter on Monday week, making next
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