Rob Page's side stormed into a two-goal lead thanks to strikes from David Brooks and Neco Williams in Cardiff. Teemu Pukki reduced the deficit just before the break, but Brennan Johnson and Daniel James netted in the second half to keep Wales' Euro dreams alive.
Bidding to reach a third consecutive European Championship, Wales will take on Poland in the play-off final at the Cardiff City Stadium on Tuesday. Poland were 5-1 winners against Estonia in their semi-final on Thursday.
"We got off to the best possible start. Everyone put in a performance. It could have been five or six in the end, but the message is that this is half-time," Page said. "We're going to enjoy the win and take this into the final. We've now got a massive effort to go. We can't wait. Roll on Tuesday."