Assat and Vaasan Sport had already sealed their Quarter-Final berths with at least one game to spare, leaving only two spots left to be filled in the last eight line-up.
Tappara needed to break the run that had seen the home side take victory in the first four matches in their decider away at Kiekko-Espoo.
After 12 minutes, that was not on course to happen, when Heikki Liedes put K-Espoo in front on a powerplay.
Aleksi Matinmikko drew The Battle Axe level with less than four minutes remaining in the first, but their parity lasted just two minutes, as Santeri Virtanen restored the hosts' advantage.
The home crowd in Espoo thought it was their night in the second period when Peter Quenneville put them 3-1 up when Matinmikko was in the penalty box for slashing.
But the third period belonged to Tappara - Jonathan Andersson reduced the deficit when 10 minutes to play, before Julius Mattila sent the match to overtime with 01:56 left on the clock.
The momentum firmly with them, Tappara completed the turnaround to win 4-3 seven minutes and 30 seconds into overtime, through Otto Rauhala's season-extending goal.
The 2024 champions will now face Tampere rivals Ilves in the Quarter-Finals.
Joining Tappara in the last eight are SaiPa, who also needed overtime to see off the lowest seeds, TPS Turku, 3-2.
SaiPa had been living dangerously throughout the series, going 2-0 down before coming back to level things up with a narrow win on Saturday to force a decider.
Though Juuso Heikkila gave them the lead inside three minutes, the nerves were not about to get a night off, as TPS hit back less than a minute later through Viljami Marjala.
Marjala was again the man to respond for Turku after SaiPa had gone in front, equalising five minutes after Mikko Petman had made it 2-1. No goals in the third period meant for overtime.
It took nearly 14 minutes of the extra period to arrive, but finally the winning goal came, and Petman was the man to set up his brother, Ville Petman, to put their hometown club into the Quarter-Finals.
In the next round, the team from Lappeenranta will take on Helsinki's HIFK.