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The visitors were still missing injured quarterback Tony Romo, without whom they have failed to win so far this season, although he is expected to feature for the first time since week two when his side take on the Miami Dolphins next weekend.
It was a scoreless opening quarter for the hosts, but they got on the board right at the start of the second when Connor Barth converted a 52-yard field goal.
Bailey went one better with a 53-yard effort to restore his side's lead, however, and that was how it stayed for the remainder of the half as both offenses struggled to make any serious ground.
Winston was intercepted for the first time in five matches by Jeff Heath towards the end of the second quarter, but the pick came deep in Dallas territory and they were unable to take advantage of the turnover.
Heath picked Winston off again in the fourth quarter, but once again the rookie quarterback was not fully punished as the Cowboys failed to add to their lead.
Indeed, it was Winston who would be the hero in the end as he ran in from a yard out with less than one minute remaining to hand his side a victory that takes them to 4-5 for the season.
The Cowboys, on the other hand, drop to 2-7 having failed to win since back-to-back victories at the start of the campaign.