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The Dolphins' comeback continued in the second half when Mike Wallace broke off a tackle to keep hold of a pass from Tannehill and force his way into the endzone.
The home side took the lead for the first time early in the fourth quarter when Lamar Miller carried the ball over from a couple of yards out.
Minnesota soon responded, though, with Blair Walsh kicking a short field goal to cut the Dolphins' lead to a single point.
Wallace thought that he had sealed the win when he added his second touchdown of the game, but Jarius Wright just kept hold of a low pass from Bridgewater to give the Vikings a chance.
Minnesota went for the two-point conversion, with Asiata carrying the ball over to tie the scores inside the final five minutes.
On the kickoff, rookie Jarvis Landry attempted to return the ball for the Dolphins, but he fumbled to give possession back to the Vikings deep in Miami territory, leaving Asiata to run it in from five yards out, putting the NFC North side back in front.
Despite results elsewhere eliminating them from playoff contention, the Dolphins continued to press, and tied the scores with just over a minute to play when Tannehill flung the ball to Damian Williams.
It looked like the game was heading for overtime, but Terrence Fede blocked a Minnesota punt in the endzone with just 45 seconds left on the clock, forcing a safety for the Dolphins to seal a home win.
The win moves the Dolphins to an 8-7 record, while the Vikings slip to 6-9.