A run of four successive wins had seen SaiPa climb above the likes of JYP to go second, but the Jyvaskyla side proved they were still a force to be reckoned with when Viljami Nieminen gave them the lead in the 16th minute.
SaiPa survived a JYP powerplay late in the first period for too many men on the ice, and they were rewarded for that defensive resilience early in the second, when Miikka Salomaki found an equaliser.
The home side had four powerplays to JYP's one over the rest of an ill-tempered second period, yet it was the visitors who scored the third goal of the game - Pekka Jormakka put JYP 2-1 up with just over two minutes to play, after Leevi Tukiainen had already spurned a penalty chance.
But with just under 10 minutes left in regulation, Samuli Niinisaari produced a second equaliser for the Lapeenranta side to force the match into overtime.
Having been one of his side's heroes earlier on, Salomaki was penalised for tripping three minutes into the extra period, and that gave JYP the chance they needed to win the match 3-2, as Jere Lassila struck the winning goal for his second point of the night.
SaiPa are now a point behind Tappara, but have played three more matches than the leaders. JYP, now on a three-match winning run, move into third place, four points behind SaiPa but with four games in hand.
Tonight's two other matches also went to overtime - Hameenlinna took the lead three times against Kiekko-Espoo, with the third a Jesse Kiiskinen goal 54 seconds into the extra period to win it 3-2. It's a fourth win running for 12-placed HPK, and a first loss in three for ninth-placed KE.
Last-placed Vaasan Sport had hope of securing a rare pair of successive wins when Theodor Johnsson put them 3-2 up with nine minutes to go away at IFK Helsinki, only for a Daniel Makiaho penalty to force OT, before Kalle Ervasti won it 4-3 for the hosts, who climb to 10th.