The champions were caught napping in the first period as the home side netted three times without reply, with two strikes from Jesse Gimblett and one from Mark Richardson.
But the fightback began after the break thanks to Craig Peacock and Daymen Rycroft, leaving the Giants with a chance to get level in the match.
The third period was a tense affair - both teams cancelled each other out for the majority of the 20 minutes until a breakthrough came with only 2:33 remaining on the clock.
Colin Shields capitalised on great work from Greg Stewart to level the game.
A scoreless five minutes of overtime ensued before a penalty shootout was played, and after Craig Peacock and Jeff Mason saw their goals cancelled out by Cardiff's Stuart MacRae and Max Birbraer, it went to sudden death.
Peacock scored once again for Belfast and Garrett Zemlak saved Stuart MacRae's shot to complete an incredible turnaround and give the Giants two points in Cardiff.
And the win could prove vital in the race for the Elite League title after Sheffield beat leaders Nottingham 5-2, closing the gap at the top to just two points.