Todd will be added to the Indians' 40-man roster and will report to Triple-A Columbus in the next few days.
The 23-year-old was 4-2 with 24 saves and a 2.20 ERA in 41 appearances for Triple-A Memphis. His 24 saves led the Pacific Coast League and he struck out 59 batters in 49 innings. He also made his Major League debut on June 4 with the Cardinals, allowing two runs on three hits over 1 2/3 innings against the Rockies.
Indians assistant GM Chris Antonetti said, "In acquiring Jess, we feel we've added an upper level pitcher who adds immediate depth to our bullpen alternatives. With his ability and track record of success, we feel that he has the potential to become a valuable member of our major league bullpen moving forward."
Todd, who was a second-round Draft pick in 2007, was ranked as the Cardinals' No. 4 prospect before the season, according to Baseball America.
Rhett Bollinger is a reporter for MLB.com This story was not subject to the approval of Major League Baseball or its clubs.