However, the point they earned turned out to be enough to win the FIFA Series New Zealand, after the hosts won 4-1 against Chile later in the evening.
Finland topped the four-team group with four points from two matches, ahead of New Zealand and Chile on three each, with Cape Verde on two.
Jacob Friis made four changes to the side that beat the All Whites 2-0 three days earlier, but it was two who retained their places who combined to score deep into first-half stoppage time, as Leo Walta cut in from the right with pace before cutting the ball across for Naatan Skytta to turn home.
It was a first international goal for the Kaiserslautern midfielder on his fourth appearance for the Eagle Owls.
Finland came close to doubling their lead early in the second half when Topi Keskinen fought hard to create shooting space at the edge of the area, before a low strike struck the base of the post and away, denying him a first international goal.
With a quarter of the match to go, Finland were pegged back by a player with a Finnish connection, as Sidny Lopes Cabral teed up SJK Seinajoki's own Kelvin Pires to equalise. Just like Skytta, it was a first goal on his fourth cap for the defender.
After 90 minutes, the match went straight to penalties - Lopes Cabral and Walta both scored before the Finns gained the upper hand as Jovane missed and Onni Valakari scored.
But from then on, nothing went right for Finland, for whom Keskinen and Jaakko Oskanen both failed to convert, as the Blue Sharks scored their next three penalties to win the shootout 4-2.
New Zealand's victory later on confirmed Finland as winners of a tournament where they remained unbeaten in 90 minutes against two sides heading for the World Cup. Friis' side will be back in action in late May and early June with friendlies against Germany and Hungary.