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With half time approaching, Tom Brady, who had been sacked four times in the first half, including one which forced a fumble, responded with a short touchdown run to put New England back in front.
The Patriots then increased that lead to 13-3 after Gostkowski kicked a career-long 57-yard field goal, which also matched Adam Vinatieri's longest in franchise history.
Soon after the break Brady found Dion Lewis with a short pass before the running-back weaved his way into the endzone.
Bailey then reduced the arrears to 20-6 with another field goal as Dallas tried to stay in touch.
However, Edelman broke through Cowboys coverage for a 59-yard TD pass, which put the Pats in total control.
Gostkowski then added another field goal as the Patriots extended their unbeaten start to the regular season.