The team said on its website the Rams outbid Miami for the 25-year-old Cook, who played for St. Louis coach Jeff Fisher in Tennessee. The Titans drafted Cook in the third round in 2009.
The Rams also have Lance Kendricks at tight end. Kendricks was selected in the second round of the 2011 draft.
St. Louis also re-signed defensive end William Hayes, who had a career-best seven sacks without a start last season. Hayes totaled eight sacks in his first four seasons in Tennessee.
The Titans declined to place the franchise tag on the 6-foot-5, 248-pound Cook, who had wanted to be paid like a wide receiver. He had 44 catches with an 11.9-yard average and four touchdowns last year when he was on injured reserve the last three games with a torn right rotator cuff.
St. Louis was 23rd in total offense last year, averaging 18.7 points.