A disappointing 4-1 loss to Slovakia in their opener had put all the pressure on Finland to keep their Quarter-final hopes alive, and having survived an early Swedish powerplay for cross-checking by Erik Haula, the Finns took a much-needed lead in the eighth minute.
Kaapo Kakko lost possession when trying to drive towards the net from the right side, but the pucked was knocked into the path of Nikolas Matinpalo on the left, who sent a powerful one-timer in off the post from distance.
Both sides then failed to take advantage of their opponents being sent to the penalty box, yet no sooner had Sweden regained full strength after a hooking call against Victor Hedman, they found themselves 2-0 down.
Inside the final five minutes of the first period, Eetu Luostarinen maintained possession well before firing towards goal. A deflection sent the puck into the air, and Luostarinen's Florida Panthers teammate Anton Lundell reacted quickest to poke it past Filip Gustavsson.
The goal was reviewed, but the officials determined Lundell's stick wasn't high, and therefore the goal stood. 2-0 was how it stayed until the end of the first, despite Sweden edging the shots on goal, 10-9.
Sweden coasted past the hosts Italy in their first match, and their chances of making it two from two increased early in the second period, when Mikko Rantanen was called for slashing, and Sam Hallam's side finally made full use of a powerplay at the third attempt.
A neat reverse pass from William Nylander found Rasmus Dahlin in space, and he hammered the puck first time past Juuse Saros with 04:39 gone in the period.
Nylander continued to be the driving force in attack for Tre Kronor, striking the post midway through the match after the Swedish defence stood firm following a tripping penalty against Joel Eriksson Ek. However, the match quickly swung back in Finland's favour.
The Leijonat were defending deep having lost Kakko for holding, and a long clearance was picked up by the alert Haula, who was pinned against the glass by three Swedish defenders, but somehow the puck deflected to the unmarked Joel Armia, who turned and beat Gustavsson to score shorthanded.
That ensured the Finns again ended a period with a two-goal lead, and much like the end of period one, the 40-minute mark arrived with tempers flaring on both sides of the ice, resulting in three players per side being penalised.
The opening minutes of the third period were a more sedate affair, as Finland created little on its first powerplay, before the Swedes peppered Saros' goal with strikes from Dahlin and Adrian Kempe after Lundell was called for holding the stick, but the Finnish goaltender was equal to everything.
Saros was beaten by another deflected Dahlin strike with nine minutes to play, and the puck was millimetres from trickling over the line, only for Lundell to flick it away just before it crossed the line.
Sweden then had another powerplay with eight minutes to go, as they dominated the shots on goal in the third period, yet they were struggling to carve out more chances, and they had to rely on Gustafsson, who made a key save to deny Lundell on the breakaway with five minutes left.
Gustafsson was pulled inside the final three minutes, but the extra attacker made no difference, and when the Swedes lost possession in the final minute, Rantanen struck an empty-netter from inside his own zone to wrap up a 4-1 win.
The other match in the group finished Slovakia 3-2 Italy, which sends the Slovaks top of the group, with Finland a Sweden level with one match each left to play, while the hosts sit bottom and now cannot progress to the last eight.