Usually they don't even ask for a boarding pass in passport control in HEL and even if they do, I think you can just explain that your visa is about to expire. In that case they may even come up with some rule that allows you to stay (or them to grant permission to stay) inside the border for an extra day but of course that is not guaranteed. They do ask for a boarding pass in security, but you don't have to pass one if you head straight to passport control.

Finnish civil servants, including border guards, usually go strictly by the book but on the other hand the penalties for minor violations in Finland are not harsh at all (if there is any) I remember seeing in Finnish airport documentary series a Russian lady who got caught having a visa that was granted on false grounds. Since there were not flights back in that day any more, they just confiscated her passport, told her to get a ticket to morning flight, get a hotel room and report back next morning. Next morning they escorted her to the gate and gave her passport back and said her current visa is cancelled but she is not banned from Finland or Schengen and she is free to apply for a new visa with correct information. I would imagine that in many other countries she would have spent a night in a holding cell.