Goals from two of their England-based contingent put Sweden in control, with Sunderland's Sebastian Larsson and Blackburn Rovers' Martin Olsson netting early in each half. Joona Toivio, who plays in Sweden for Djurgarden, halved the deficit 18 minutes from time to set up a close finale, but the hosts could not find an equaliser.
The win lifted the visitors onto 21 points and kept them in contention for the one direct qualification spot available to the best runners-up. Third-placed Hungary, not in action tonight, can still finish level on points with Sweden, but Erik Hamren's side have a better head-to-head record.
Swedish nerves settled quickly as Larsson put them ahead in the eighth minute. The midfielder swept a low shot into the left corner from 12 yards after being picked out by Kim Kallstrom's right-wing cross.
Olsson extends lead
Their Scandinavian rivals came back at them, Teemu Pukki seeing his long-range effort tipped over the bar by Andreas Isaksson, while the goalkeeper saved with his feet to deny Kasper Hamalainen minutes later.
At the other end, the visitors threatened a second when Daniel Majstorovic met Rasmus Elm's corner, but the defender's header went wide. Finland's hopes of qualification disappeared long ago, but they were in no mood to lie down, Mika Vayrynen heading over in the opening minutes of the second half.
Sweden established a two-goal cushion in the 52nd minute, though, as Olsson turned the ball home from close range after Zlatan Ibrahimovic had gone close from Larsson's cross. The hosts pulled one back in the 73rd minute when Toivio converted Roman Eremenko's ball in, but Sweden held on for a crucial win.