Wilson was presented with a glorious opportunity after 25 minutes to hand Wales a morale-boosting victory ahead of their World Cup qualifiers against Belarus and Estonia.
Brennan Johnson's trickery had created the chance but Wilson's kick was too close to Finland goalkeeper Carljohan Eriksson, who guessed correctly to his right on his debut.
Substitute Rubin Colwill also spurned a close-range opportunity and not even the late arrival of Real Madrid's Gareth Bale could reward Wales' increasing superiority.
Wales had been hit by a raft of withdrawals ahead of their second visit to Helsinki in the space of 12 months.
Aaron Ramsey, Joe Rodon, David Brooks and Neco Williams were among several players ruled out by injury, while goalkeeper Adam Davies had tested positive for coronavirus.
Kieffer Moore, who scored the late winner when Wales won a Nations League clash in Helsinki last September, was deemed a close contact of Davies and was also unavailable.
Bale was named among the substitutes, with interim boss Robert Page mindful of those two World Cup qualifiers.
Leicester goalkeeper Danny Ward - the sole survivor of Wales' 4-0 defeat to Denmark at Euro 2020 - led a starting XI possessing just 160 caps.
Johnson made his first start after two substitute appearances, with the Nottingham Forest youngster operating in an advanced role on the right of a 3-4-3 formation.
Finland also fielded an experimental side and Wales settled the quicker with Wilson, sent off on his last appearance for the national team just over two months ago, particularly prominent.
Ethan Ampadu headed over a Dylan Levitt free-kick and Wilson saw an effort blocked inside the opening 10 minutes.
Wilson soon put the ball in the net but the assistant referee's flag was up for offside against Johnson.
Both goalkeepers were being well protected but that was to change when Johnson induced a rash challenge from Niko Hamalainen.
Wilson stepped forward but, just like Bale had done against Turkey at the Euros, the Fulham forward failed to convert from 12 yards.
The tempo dropped as half-time approached and Page turned to his bench at the interval.
Goalkeeper Wayne Hennessey replaced Ward to win his 97th cap and fellow substitute Ben Davies took the armband.
Wilson saw another shot blocked but Finland began to show more ambition, with Tom Lockyer and Levitt called upon to deflect efforts wide of Hennessey's goal.
Tyler Roberts' endeavour won Wales a free-kick which signalled danger as Wilson has developed quite a scoring reputation from outside the penalty area.
Wilson beat the wall but not the diving Eriksson, who comfortably pushed the ball to safety.
Eriksson was grateful moments later to see Matt Smith's low drive skid just wide of his far post.
The numerous changes seemed to work in Wales' favour as the fluency Finland had showed at the start of the second half escaped them.
Wales were also galvanised by the arrival of Bale, who had two speculative attempts in the brief period he was on the pitch, but they could not find a breakthrough.