The Lions took the lead right at the end of the first quarter as Stafford zipped a pass to Calvin Johnson in the endzone to complete a two-yard touchdown.
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Dan Bailey got the Cowboys on the scoreboard with a 53-yard field goal midway through the second quarter, and it was the visitors who took a lead into the break courtesy of a score with just 51 seconds left of the half.
Bailey hit his second 53-yard field goal of the game to extend the lead late in the third quarter, but it was cut back to three points at the start of the fourth when David Akers converted a kick from 20 yards.
Detroit hit back midway through the final quarter when Joique Bell powered over from a yard out, but Bryant claimed his second touchdown of the day when he broke through a tackle before sprinting down the touchline for a 50-yard score.
The Lions made it a three-point game once again late in the fourth quarter when Reggie Bush reached the ball over the line from a yard out, but the Cowboys extended the lead to six with 1:07 left in the match through another Bailey field goal.
Stafford capped off a high-scoring final quarter in dramatic fashion, however, as he tunnelled in from a yard out to complete an 80-yard drive in under one minute with the crucial touchdown.
Akers added the extra point with time running out as the Lions secured a last-gasp victory to move to 5-3 for the season.