The deal was announced Friday.
The 26-year-old Grichuk hit .238 last season with 22 home runs and 59 RBIs in 122 games. He became expendable after St. Louis got big-hitting left fielder Marcell Ozuna in a trade with Miami last month, and because of the emergence of Tommy Pham in center.
In return, the Cardinals receive a much-needed boost to their bullpen in Leone. The 26-year-old right-hander was 3-0 with a 2.56 ERA in 65 games last year for Toronto, striking out 81 in 70 1/3 innings.
Greene was a seventh-round draft pick of the Blue Jays in 2013 and went 5-10 with a 5.29 ERA at Double-A New Hampshire last season.
