Varvaro was 3-3 with a 2.63 ERA for the Braves last year, his fifth season in the majors. The Red Sox also sent cash to the Braves.
In 61 relief outings last season, the 30-year-old Varvaro had 50 strikeouts and 13 walks. The Braves designated him for assignment on Monday.
Kurcz, 24, was 3-2 with a 2.14 ERA with Double-A Portland last season.