Chicago and Atlanta also swapped four international pool slots in Sunday's deal, with three going to the Braves and one to the Cubs.
The 25-year-old La Stella made his major league debut this year and hit .251 with one homer and 31 RBIs in 93 games. The lefty batter was selected by Atlanta in the eighth round of the 2011 draft.
The 24-year-old Vizcaino begins his second stint with the Braves. After being acquired from the Yankees in 2009, Vizcaino made his debut with Atlanta in 2011. In 2012, he had Tommy John surgery and was traded to the Cubs.
The hard-throwing Vizcaino appeared in five games with Chicago in September and allowed three runs in five innings.