"It's always difficult playing second throughout the weekend," man-of-the-match Barry told Sky Sports.
"(United) obviously put a bit of pressure on, any away game in the Premier League is really tough. But to come to Villa Park is a massive three points for us, we are delighted today.
"We've got to back ourselves now although they are a club who have won many championships recently. We are a team full of experience and have players that have won championships elsewhere. We know what it is all about."
Mancini, whose side went into to a league game not as leaders for the first time since October, was thrilled.
"This evening was a difficult game, we knew this before the game but I think the guys played fantastic football," he said.
Marc Albrighton of Aston Villa and Adam Johnson of Manchester City look on as a streaker runs on to the pitch during their match on Sunday
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