The hosts scored twice in the opening six minutes as they eased to a win that keeps them in touch with the Champions League chasing sides.
The Sweden international striker was rested ahead of the Champions League clash with Chelsea and his absence was felt as the hosts failed to find a way through.
The Juventus midfielder struck twice, while Jorge Valdívia and Alexis Sanchez had efforts contentiously ruled out for offside in the second half.
Neymar netted the equaliser and then set up the winner as Dunga's men avoided a shock result in their first game of the Copa America.
The Azzurri conceded just seconds after having a goal disallowed but the Lazio man's delightful penalty ensured a draw after the hosts missed one of their own.