Vazquez and reliever Boone Logan were sent to the Braves for four prospects, the White Sox said.
Vazquez, 32, who will be joining his fifth major league team, was 12-16 with a 4.57 earned run average for the White Sox last season. The right-hander has a career record of 127-129 with a 4.32 ERA in 355 games with the Montreal Expos, New York Yankees, Arizona Diamondbacks and White Sox.
Logan, a left-hander, has pitched only for the White Sox, compiling a 4-4 record with a 5.87 ERA in 144 games over three seasons.
Atlanta sends catcher Tyler Flowers, left-handed pitcher Santos Rodriguez and infielders Brent Lillibridge and Jonathan Gilmore to Chicago.
Flowers batted .288 at Class-A Myrtle Beach last season and was named to the Carolina League all-star team.
Rodriguez played for the Gulf Coast League Braves, winning one game and losing two with an ERA of 2.79.
Lillibridge spent most of the season with Triple-A Richmond and Gilmore played in the Appalachian League, making its all-star squad.