KuPS' 6-2 win over Gnistan in midweek counted for nothing in the opening exchanges of their trip north, falling behind five minutes in when Tuomas Kaukua's deflected shot looped over the goalkeeper and in.
The champions needed a huge slice of luck to restore parity after the break - goalkeeper William Eskelinen's clearance struck Otto Ruoppi on the back and fell kindly for Mohamed Toure, who lofted the ball past the stranded shot-stopper.
Having known nothing about his assist, Ruoppi knew plenty about the goal that put KuPS in front on 65 minutes - he made space for himself in the box to meet Saku Savolainen's low cross and finish well.
However, Oulu were not about to lie down - 12 minutes from time, a Julius Korkko corner was nodded home by Miika Koskela to secure an unlikely point and lift the hosts off the bottom of the table for the first time since March.
The side from Kuopio have remained one point clear at the top of the standings after Ilves were held to a 1-1 draw away at SJK on Saturday.
SJK were going to provide tough opposition in Seinajoki on a four-match unbeaten run, and that proved to be the case when Samuel Chukwudi put them ahead after 14 minutes.
But the Lynx, in good form themselves, got themselves on level terms on 64 minutes when Jardell Kanga converted from the penalty spot, shortly after coming on.
It's now just one league loss in the last eight for Ilves, while SJK drop a place to fifth.
SJK's fall in the Veikkausliiga table was due to HJK picking up a 2-0 win away at VPS to put an end to their two-match losing run.
The victory was down to Kevin Kouassivi-Benissan, recently returning to the side after a loan at Kosice in Slovakia.
The 26-year-old came on at half-time and took nine minutes to open the scoring for the Helsinki side - aided by a Teemu Pukki assist - before he sealed the victory with 20 minutes to play when in the right place to tap home a rebound.
HJK are fourth but only four points behind KuPS and eight clear of VPS, who sit sixth after three losses on the spin.
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