It was the second big play for Hunt, who also dropped a running back for a loss on a third-and-1.
The Dolphins offense, meanwhile, couldn't manage a first down in the first quarter while Daunte Culpepper was sacked twice and was called for intentional grounding on another play that would have resulted in a sack.
But the offense came to life on the fourth drive with a beautiful march that ended with Sammy Morris' 2-yard touchdown dive that gave Miami a 7-3 lead.
Culpepper finished 7-for-9 for 86 yards on the night.
The Dolphins, meanwhile, shut down Tampa Bay the rest of the first half after giving up a field goal on the Bucs' first drive.
But the pass defense had its problems in the second half.
There was another breakdown — similar to what happened against Jacksonville — when Edell Shepherd got behind rookie Jason Allen, who played cornerback against the Bucs, and safety Deke Cooper and caught a 48-yard touchdown pass from Gradkowski to close the Dolphins' 13-3 lead midway through the fourth quarter.
After the play, Coach Nick Saban was caught on camera yelling at Allen.
Earlier in the second half, Gradkowski connected with Maurice Stovall for a 36-yard pass to the Dolphins 19-yard line. But on the next play, Vickerson recovered a fumble to end the scoring threat.
It was, all in all, an encouraging performance by the Dolphins after a forgettable start. And the best news was that it didn't appear that anybody sustained a significant injury.