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Miami's lead was short-lived, though, as Smith handed the ball off to Chris Owusu, and the rookie wide receiver found the endzone from 22 yards out with his first rushing play in the NFL.
The Jets quarterback lost the ball on the next drive for the visitors though, leaving Daniel Thomas to power over for a Miami touchdown.
Chris Ivory gave the lead back to the Dolphins after Smith found him with a short pass out wide, but Charles Clay got free in the final minute of the second quarter to make sure that the home side were ahead at the break.
The visitors were quick to respond though, as Eric Decker caught a Smith pass before running from midfield for a 73-yard Jets score.
The Jets were able to take the lead for the first time in the second half when Folk made a pair of field goals, tying the game before edging the visitors in front early in the fourth quarter.
Smith put the win beyond doubt when he found Jeff Cumberland with a deep throw with just over five minutes to play, while another Folk field goal late on made the margin more comfortable.
Both sides had already been eliminated from playoff contention ahead of the start of play, and the defeat leaves the Dolphins third in the division, while the Jets finish bottom with a 4-12 record.