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Miami were unable to answer with a touchdown of their own from their first drive, but kicker Caleb Sturgis added a 48-yard field goal to get the home side on the board.
Tannehill made a strong start to the third quarter, getting the hosts into a good position with a run before picking out Jarvis Landry for the rookie wide receiver's first NFL touchdown.
The Packers responded with Rodgers scrambling behind the line of scrimmage, avoiding several attempts to sack him before firing the ball in to Randall Cobb.
However, Miami looked dangerous again at the start of the final 15 minutes, with Mike Wallace hauling in a catch on the edge of the endzone before Lamar Millar ran the ball in on the next Dolphins drive.
Green Bay trailed by four heading into the final seconds of the game, but Rodgers found Nelson to help get the NFC North side into the red zone.
With just six seconds remaining on the clock, the Packers quarterback fired a short pass to tight end Quarless to edge the visitors to the victory.
The result leaves the Packers with a 4-2 record, while the Dolphins slip to 2-3, having already had their bye week.