Coach Roberto Mancini again named an experimental Italy side, but that mattered little as the hosts dominated from the off in Cesena, deservedly taking the lead on the half-hour mark through Inter Milan midfielder Barella.
An own goal from Gianluca Mancini did give Hungary hope of building on their surprise win over England at the weekend with a credible point in Cesena, but Italy held on to move onto four points from their two opening League A, Group Three games.
With Germany drawing 1-1 with England in Munich on Tuesday, Italy sit top of the group, having won just their fourth match from 12 since securing European Championship glory last July.
"It was a good match in the first half, but their goal put us in a bit of a tough place," Mancini told Rai Sport. "We should have won by a much a higher margin.
"We were also a bit tired, it's normal. To concede a goal in a match that should have already been finished off is difficult. We are young side, who will have to work hard. Our start has been good."