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.
The Spanish boss has suggested that he was not pleased by Neymar's showboating at the end of the fixture as Barcelona moved close to the treble.
The Uefa president has urged his organisation's 54 members to vote against the Swiss due to the latest corruption scandal to hit the governing body.
The goalkeeper has been jeered by his own supporters after making a string of high-profile mistakes this season.