A home win for Sport would see them reach the last eight with a game to spare, but they had to wait until late in the third period before ultimately sealing the 4-2 victory.
They took the lead in the first period through Linus Andersson, only for Matic Torok to draw the visitors level midway through the second.
With just over 13 minutes to play, Sebastian Stalberg regained the lead but KooKoo hit back again with Vilmos Gallo squaring the match at 2-2 inside the final six minutes.
But after Stalberg set up Viljami Nieminen to score, the side from Kouvola could not find a third equaliser and their season was ended by an empty-netter from Carl Mattsson in the last minute.
Sport move on to face second seeds Ilves in the Quarter-Finals.
Having trailed 2-0 in the series, SaiPa are now back on level terms against TPS Turku after a 2-1 win in game four.
All three goals in Turku came in the second period and it was the away side who broke the deadlock - too many men on the ice for TPS gave Santari Airola the chance to score on the powerplay.
Aarne Talvitie got the hosts back on level terms with three minutes of the second to go, only for Mikko Petman to score what turned out to be the winner for SaiPa, 21 seconds before the final break.
The series decider will be played at SaiPa's Lappeenrannan Jaahalli on Monday.
A fifth game will also be required to decide which of Tappara and Kiekko-Espoo reach the last eight, after the "battle axe" won 2-1 at home to level the series at 2-2.
The hosts had three powerplays in the first period to K-Espoo's two, but neither side could break the deadlock in the opening 20 minutes.
The first goal arrived five minutes into the second, courtesy of Daniel Brickley.
K-Espoo finally took advantage of a powerplay at the fifth attempt when Peter Quenneville scored for the second match running, but last year's champions grabbed the winner early in the final period through Benjamin Rautiainen.
A spate of late penalties hindered the Liiga newcomers chance of a late equaliser, but at least they will host the decider on Monday.
Jukurit draw level in the Playout
The Playout to decide which team retains their Liiga status next season is level at one win each, after Max Ellis was the hero for Jukurit against Pelicans.
He opened the scoring in the first period, then, after Lars Bryggman's late equaliser sent the match to overtime, Ellis scored the winner 11 minutes into the extra period, to secure a 2-1 win.
Game three of the best-of-seven series takes place on Tuesday.