Away win secures Sweden top-two spot
Hungary, in third place, cannot now catch the Swedes who have a better head-to-head record.
An early goal by Sebastian Larsson put the visitors in front but a young Finnish side pressed hard in the pouring Helsinki rain and controlled the first half without threatening keeper Andreas Isaksson's goal.
The home team fell further behind in the 52nd minute when Martin Olsson struck.
Joona Toivio pulled one back for Finland, who were routed 5-0 by Sweden earlier in the competition, after 73 minutes.
"I am very happy. Finland played better than us today. We were effective and we won," Sweden coach Erik Hamren told reporters.
His talismanic striker Zlatan Ibrahimovic was shown a yellow card and will miss the last game of the campaign against group leaders Netherlands on Tuesday.
"It was cheap. He [the referee] apologised to me and said he was wrong," said Ibrahimovic.
Netherlands, who are three points clear of Sweden, host Moldova later on Friday.
The winners of Europe's nine qualifying groups and the best second-placed team gain automatic berths in the finals alongside joint hosts Poland and Ukraine while the other eight runners-up enter two-legged play-offs for the remaining four places.