SaiPa beat Assat 4-0 in the second match of what was a three-match losing streak coming into this game for the Pori side, but revenge looked possible midway through the first half, when Valtteri Lipiainen's saved shot was tucked home on the rebound by Feetu Knihti.
Things got better for the hosts before the first period was over, after Miikka Salomaki was penalised for slashing, Peter Tiivola made use of the space inside the SaiPa zone to finish coolly on the powerplay.
That goal was timed at 18:29, and at 18:28 in the second period, Assat struck again when a stretched away defence gave Kasper Puutio too much room to fire in goal number three from the left side.
However, SaiPa gave themselves hope in the final minute of the second, as Miro Tauslahti played the puck across to Brain Kramer in space on the right, and he powered the puck across the goaltender for 3-1.
That goal gave the Lappeenranta side plenty of belief, and when Maxime Fortier worked hard in front of the net four minutes into period three, he was rewarded with a goal to make it a one-goal game.
SaiPa then survived a pair of interference calls before completing the comeback with just 11 seconds left of regulation - they threw on an extra attacker, maintained possession inside the Assat zone and found the arriving Emil Kuusla to slam the puck home and send it to overtime.
With the stuffing knocked out of the Aces, there was only going to be one winner in the extra period, and just over two minutes in, Kramer drove through the Assat zone all on his own and struck his second of the game to win it 4-3.
Assat remain in eighth place, now eight points behind Ilves in seventh and 11 points off the top four. Two more points for SaiPa is enough for them to climb to second, three points behind Tappara, as they extended the current-best winning run in the Liiga to five matches.