Having entered the qualifying process at Phase 2, Finland began their Group 2 campaign promisingly with a narrow 29-28 loss away to Montenegro in November, before being well-beaten by Hungary, 32-23.
A home win against Slovakia, who also suffered back-to-back losses before Christmas, was a must to maintain realistic hopes of a place in net January's European Championship.
Slovakia scored the first goal of the game, but would not lead again until the 23rd minute, at 9-8.
In that time, Finland twice led by a two-goal margin, but there was very little to separate the sides, as reflected in the 12-12 half-time score.
The Finns were being aided by Miska Henriksson, who scored four of his team-high five goals before the break.
The tension continued in the second half, with the lead regularly changing hands, and neither side opened up a lead by more than one goal until Fredrik Karlsson made it 21-19 to Finland in the 54th minute.
Finland failed to score after Benjamin Helander put them 22-20 to the good with a little over three minutes remaining, but despite Marko Jurkovic halving the deficit, Slovakia could not find an equaliser in the final seconds.
The win takes Finland to two points in Group 2, two behind second-placed Montenegro, who were beaten 29-26 by Hungary.
Sunday sees the reverse fixtures take place - Finland travel to Hlohovec knowing a repeat of last night's results will give them a real chance of finishing in the top two.
If they can, Ola Lindgren's side will qualify for the EHF Euro for the very first time.
