After the Saints built up a 10-3 advantage in the first half, any hopes of a Tampa Bay victory took a further slide when New Orleans added to the deficit through two late field goals.
But the veteran quarterback was determined not to let his side go down without a fight, capping off a 10-play drive with a one-yard touchdown pass to Cade Otton as the clock ticked down to three minutes.
Battling the clock, as well as a stingy Saints defence, Brady led another meticulous drive downfield before finding Rachaad White in the end zone with just three seconds left.
Kicker Ryan Succop then nailed the extra-point conversion, sealing the miraculous victory and keeping Tampa Bay's playoff hopes alive.