Clabo told the Atlanta Journal-Constitution that he received a call at 7:30 a.m. ET Thursday to inform him of his release.
"We're really disappointed, but at the same time we're excited about new experiences. We'll have to see where we go from here. This was definitely unexpected," he told the newspaper.
The Falcons currently are only about $2 million under the salary cap and will need more room than that to sign their draft picks.
Clabo's release only will free up about $900,000 in the short term, but the team designated him as a June 1 cut.
After June 1, the move will allow the Falcons to clear up about $4.5 million in cap space.
Clabo has started every game since 2008. His departure could open the door for Lamar Holmes, a third-round pick last year, to compete with Mike Johnson for a starting job. But the Falcons also could look for a replacement in the draft.
"Unfortunately, we had to make another difficult decision regarding our roster today," Falcons coach Mike Smith said in a statement. "Tyson was a valuable contributor to our football team during his time here and we appreciate everything he did for this team and in the community."