The trade was announced Sunday.
Venditte, 31, has pitched three times in spring training for Seattle this year and twice for Italy in the World Baseball Classic.
Venditte was 0-0 with a 5.73 ERA in 15 games for Toronto and the Mariners last year. Able to throw with both arms, he's 2-2 with a 4.97 ERA in 41 games in the majors.
Curletta, 23, hit a combined .251 with 17 homers and 67 RBIs in Double-A and Class A in the Dodgers' system last season. He was sent from Los Angeles to the Phillies in the trade for Carlos Ruiz.