The move leaves Luis Avilan as the only left-hander in Atlanta's bullpen.
Veal gave up one run in 1 1-3 innings in Monday night's 5-2 loss to the Phillies. He allowed four runs in 3 1-3 innings for a 10.80 ERA this season.
Cunniff opened the season with Atlanta, appearing in 10 games, and was optioned to April 21. He allowed two runs, one hit and six walks in nine innings.