The 26-year-old tight end joined the NFC West side earlier this year after the Tennessee Titans refused to offer him a deal as a wide receiver.
The Rams' head coach told The Tennessean: "He's obviously got years of experience under his belt, he understands the league, he understands defenses and he's picked things up really well.
"We're excited about creating mismatches, and offensive success these days is about creating mismatches.
"He's going to play all over the place. We've even got him in the backfield, so we've got some good things for him."
Cook recorded 523 receiving yards and scored four touchdowns for the Titans in 2012.