The Cowboys were facing a third-and-5 at their own 30 with 1:38 remaining and the game tied 19-19. Dallas had blown a 19-9 lead, with Julio Jones catching a game-tying 34-yard touchdown pass from Matt Ryan on the previous possession.
Dallas was collapsing, but Quinn gave them a chance to reset.
"I thought let's give our offense a chance. I was going into the mindset to say we would stop them," said Quinn. "And if we did, I didn't want to use that 40 seconds on the punt. We'd have a chance as opposed to allowing them to use that time. We could go ahead and use the timeout then and save the time and still have one timeout left."
Dallas quarterback Dak Prescott hit Michael Gallup for a gain of 10 to keep the possession moving. Prescott hit two more big passes on the possession, one to Ezekiel Elliott and the other to Cole Beasley to put the Cowboys in field-goal range.
Brett Maher hit the field goal from 42 yards out, and the Cowboys won 22-19 at Mercedes-Benz Stadium for their first winning streak of the season. It brings them up to 5-5 and within a game of first place in the NFC East after Washington lost to Houston.
The Falcons fall to 4-6 and saw their playoff hopes effectively end.
"I was confident as I have been every time I step on the field," said Maher. "Everyone in here is doing so much for our team for the offense to be able to drive the ball down the field and get us in position."
It is the fourth time Atlanta has lost on the last play of the game this season.
Until the Cowboys got the ball with 7:12 to go in the third quarter for their second possession of the second half, their offense had been very reliant on Prescott's arm. The Falcons led 9-6 in a game without a touchdown, but Prescott was only 10-for-17 for 104 yards.
Elliott had been fairly sheltered throughout the game, and the Falcons did well to control him when he did get the ball.
The Cowboys went on a 14-play, 75-yard drive that took 7:53 off the clock and saw Elliott touch the ball on seven of the plays, and made the Falcons respect the run for the first time all day. It set up Prescott for some play-action passes, which he used for a big play to Amari Cooper.
Prescott finished off the drive and gave the game its first touchdown with a 4-yard scamper to put the Cowboys up 12-6.
"Our O-line was being very dominant up front. Whenever those guys do that and open up those holes, it's really easy for me," said Elliott. "Those guys are the engine of this team. When they're going like that, we're going to be hard to beat."
The dam finally opened for Elliott, who padded their lead with a 23-yard touchdown run after a Ryan pass hit the hands of Calvin Ridley and bounced into the arms of Dallas linebacker Leighton Vander Esch for a critical interception.
The touchdown run from Elliott was on the second play following the interception, and gave the Cowboys a 19-9 lead with 12:33 remaining.
For the Falcons, the Ridley drop was a microcosm of the way the offense never quite found a rhythm during the game.
The opening possession of the game saw the Falcons reach the Dallas 38 but fail to get even a field goal attempt thanks to Demarcus Lawrence pushing Jake Matthews into Ryan for a drive-ending sack.
A miscommunication between Ryan and Mohamed Sanu on third-and-4 at the Dallas 24 led to Atlanta having to settle for a field goal in the second quarter.
Another Lawrence sack on a third-down play knocked the Falcons out of field-goal range. Ryan dunked a pass behind Austin Hooper late in the first half that would've been close to a first down that forced another field goal for the Falcons.
"I thought our offensive line did a nice job in the second half adjusting to what Dallas was doing. I thought they played well and battled. Across the board, we just needed to play a little bit better in the first half," said Ryan. "I think if we were able to convert a few of those third downs and keep some drives going, that could have been a different first half."
Ryan led the offense over the 50 on seven of their eight possessions, but they only made it into the end zone on the Jones touchdown.
Both teams will have a quick turnaround heading into games on Thanksgiving. The Cowboys will return home to host the Redskins in a huge NFC East showdown, while the Falcons will head to New Orleans to face NFC South-leading Saints.