First of all, this is only a good argument if "extending the game" is what you want. Extending the game isn't always the best option. Sometimes gambling gives you a higher percentage of winning than playing it safe.
Secondly, this isn't even true. The game isn't over if the Falcons fail. The stakes are essentially the same: stop the Saints from getting a first down (or two) or lose.
Let's look at the three possible outcomes:
1) Convert the first down. Ball at around midfield, you have 3 timeouts (with the 2 min warning), and a "clutch qb" at home. You're in the driver's seat. You have to get 15 yards for a pretty good look at a tie, and you have a decent shot at just winning outright.
2) Fail the conversion. Ball at around the ATL 43 for the Saints.
a) If you stop them, they have to convert a long field goal and risk giving you good field position, or punt and try to pin you deep. You'll get the ball anywhere from the 1-20 probably.
b) If you don't stop them, you're screwed.
3) Punt. Assuming a good punt, you'll probably put them around the 15.
a) If you stop them, they have to punt. You'll probably end up with the ball around the 30-50.
b) if you don't stop them, you're screwed.
b) is the crucial part. It almost doesn't matter if you punt - you have to stop them either way, and even if you get the ball back you're looking at (I think) 1:20 of time with no timeouts. the difference is just field position.