Barella, 22, gave Italy the perfect start by scoring with a deflected shot after seven minutes and Kean, the first player born after 2000 to play for Italy, added the second in the 74th minute.
Italy's first home win since they beat Israel in a World Cup qualifier in September, 2017 gave them an ideal start as they attempt to put their failure to qualify for last year's World Cup behind them.
Finland clearly set out to frustrate Italy but that plan had to be scrapped when a clearance went straight to Barella who met it first time from the edge of the area and a deflection wrong-footed goalkeeper Lukas Hradecky.
The visitors came into the game but made little headway and Italy's Federico Bernardeschi had a penalty appeal waved away before halftime.
Italy survived a major fright in the 65th minute as Robin Lod burst down the right in acres of space and slipped the ball to Teemu Pukki who turned his shot wide from six metres.
Having got into plenty of good positions only to be let down by their final pass, Italy finally got it right when Ciro Immobile slipped the ball through to Kean and the 19-year-old fired a confident finish past Hradecky.
Serie A's leading scorer Fabio Quagliarella was brought on and the 36-year-old fired a right foot shot against the crossbar.