There was a touchdown before the end of the first half, however, as Christian Ponder took advantage of some good blocking to dive into the end zone after a six-yard run, giving the Vikings the lead.
Their four-point advantage did not last long in the third quarter, however, as the Cowboys scored early on.
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The visitors responded to the Cowboys' two touchdowns in quick succession with a score of their own as Ponder found Kyle Rudolph, who somehow eluded a number of tackles on his way into the endzone.
The Vikings then regained the lead with 5:49 left in the fourth quarter when Adrian Peterson showed incredible strength to power his way over the line from 11 yards, although Walsh put his extra-point kick wide.
It was Dallas who were to have the last laugh, however, as Romo's pass found Dwayne Harris for a seven-yard score with just 42 seconds left on the clock to clinch the win for his side.
The result means that the Vikings are still without a win on their travels this season as they stay bottom of the NFC North, while the Cowboys extend their lead in the NFC East.