Barnett was waived by the Philadelphia Eagles on Friday after being selected by the team in the first round of the 2017 draft.
In seven seasons with the Eagles he accumulated 21.5 sacks, 76 quarterback hits and 37 tackles for loss.
Barnett's best season was in 2019, when he registered 6.5 sacks and 22 quarterback hits with 10 tackles for loss.
In Week 1 of the 2022 season, Barnett suffered a season-ending torn ACL against the Detroit Lions.
This season, Barnett has three tackles and zero sacks and played in only 20% of Philadelphia's defensive snaps, the second lowest of his career.
Houston had higher waiver priority than the Dolphins, who would have reunited Barnett with defensive coordinator Vic Fangio and added some depth after they lost Jaelan Phillips to an Achilles injury.