The Mystics, seeking their third consecutive win, looked to have the match wrapped up after fighting back from three goals down to take the lead at 50-49 with nine seconds to go in regulation time.
But with a couple of seconds to go, defender Vilimaina Davu was penalised in the shooting circle and Australian international Catherine Cox calmly slotted the equaliser to extend the game by another 14 minutes.
The Mystics, coming off two consecutive wins over New Zealand teams, Central Pulse and Canterbury Tactix, looked a much better side than their eighth placing in the standings.
They more than matched the fifth placed Swifts who were coming off a bye.
Circle defenders Davu and Leana de Bruin as well as wing defence Sheryl Scanlan gave Cox and Susan Pratley little space to operate.
But at the other end, the Mystics sorely missed the presence of goal attack Paula Griffin, sidelined after a tonsil operation.
Coach Yvonne Willering gave Grace Rasmussen her league debut and the young goal attack looked a little out of her depth at times.
Many a time when a shot was on, she showed a lack of confidence, her extra pass to get close for the shot often foiled by an alert Swifts defence.
The first quarter was shared 12-12 after both sides cancelled each other's turnovers out.
But the Mystics, finding their rhythm behind a tight defence, outscored the Swifts 13-10 to take a deserved 25-22 halftime lead.
Maintaining their passing accuracy, they shot out to a six-goal lead midway in the first quarter as the Swifts struggled to keep pace.
But a turnover gave the Swifts the chance to get on a mini-roll and cut the gap to three, 35-38 at the end of the third quarter.
The Mystics were slow out of the blocks in the final quarter which turned into a see-sawing affair.
The scores were tied at 41-41 before the Swifts jumped out to a 48-45 lead.
Led by their indomitable captain Temepara George, the Mystics fought back to 49-49 with 19 seconds to go.
The first period of extra time saw the teams separated by just one point with the Swifts taking a 57-56 lead but then a wild pass by the Mystics gave the Swifts the chance to seal the match 64-62.
New South Wales Swifts 63 (Catherine Cox 35/39, Susan Pratley 28/28) Northern Mystics 62 (Catherine Latu 48/52, Grace Rasmussen 14/19). Fulltime: 50-50. Halftime 22-25.
