The Yankees received cash in Wednesday's deal.
The 28-year-old right-hander was 3-1 with a 5.96 ERA in 25 games before the Yankees designated him for assignment June 16.
Opposing hitters were batting .235 against him; right-handed hitters had a .180 average. Veras had allowed no runs in nine of his final 12 appearances.
He has an 8-4 career record with three saves and a 4.43 ERA.
Cleveland must make a roster move before adding Veras to its 25-man roster.
The bullpen has been a major worry to manager Eric Wedge all season, converting only 13 saves in 26 opportunities.
During a recent six-game losing streak, the Indians twice blew leads of five runs and another of seven runs, and the bullpen gave up four runs in the ninth inning of a 5-4 victory over Pittsburgh on Tuesday in which the Pirates had the potential winning run on base.
Last season, Veras was 5-3 with a 3.59 ERA in 60 appearances and was second on the Yankees in relief innings pitched and appearances.