The hosts had trailed 4-0 at half-time as Kastriot Kastrati, a Tim Varynen brace and Moshtagh Yaghoubi gave the visitors an unassailable lead.
Emyr Huws pulled a goal back for the Young Dragons from the spot, but Varynen completed his hat-trick to compound Welsh misery.
Geraint Williams' side had started their campaign with a 1-0 win over Moldova, and had hoped to add to their tally with Group One heavyweights England yet to get under way.
But any hopes of chalking up a second consecutive win were dashed inside 11 minutes.
Firstly, Kastrati opened the scoring from Yaghoubi's corner in the seventh minute before Varynen tapped in a simple finish to double the visitors' advantage.
Midfielder Lee Lucas had a couple of half chances as the home side looked to get back in the game, but they fell further behind on the half-hour as Varynen coolly made it 3-0.
Yaghoubi added the fourth six minutes before the break to leave Wales with no hope of recovery.
They did at least pull a goal back as midfielder Huws slotted a penalty after Wes Burns was fouled in the box.
But Finland rounded off the scoring to cap an emphatic win.