The 4-1 defeat to Finland 24 hours earlier had put Sweden at real risk of missing the Quarter-finals, and would need a victory over Slovakia, who had already been the Finns by the same scoreline earlier in the competition.
But the Swedes came out of the blocks quickly and were leading the shot count 4-1, only to suffer a setback when they were guilty of having too many men on the ice.
Yet, the first powerplay going Slovakia's way proved no obstacle, as Joel Eriksson Ek attacked from his own zone and found Adrian Kempe, whose shot was saved by Samuel Hlavaj, only for Eriksson Ek to force the puck home on the rebound to score shorthanded in the eighth minute.
Going behind seemed to spark the Slovaks into life - Oliver Okuliar, who plays his domestic hockey in Sweden, struck the post seconds later, before a long period of attack resulted in Simon Nemec finding Juraj Stavkovsky wide on the right, and he hammered the puck past Jacob Markstrom despite the tight angle.
Tomas Tatar and Rasmus Dahlin conceded a penalty apiece before the first period was out, but neither side could make use of their respective powerplay, though Slovakia levelled the shot count at 13 each despite being 7-1 down after the Swedish goal.
Sam Hallam's side were under pressure early in the second when Lucas Raymond was called for hooking, put the Swedes withstood another Slovakia powerplay, before Pavol Regenda was sent to the box for kneeing Oliver Ekman-Larsson.
That was costly for the Central Europeans, as Sweden scored just four seconds into the powerplay. The face-off was worked back to Dahlin, who squared to Kempe on the right, who slapped the puck first time beyond Hlavaj.
However, the Swedish lead again didn't last long - Tre Kronor had just about survived a powerplay after high-sticking from William Nylander, but Slovakia kept pouring forward and when Peter Ceresnak's shot from range was blocked, Martin Gernat made no mistake in powering in the rebound to make it 2-2 just before the midpoint.
As the second period wore on, momentum appeared to be swinging back towards Sweden, who took advantage with just over five minutes left, as Filip Forsberg drove through the middle and teed up Elias Pettersson to place the puck past the Slovakia goaltender.
This time, however, there would be no Slovak equaliser, as though the Swedes failed to capitalise on powerplays either side of the second interval, they opened the first two-goal cushion with 12 minutes to play, as Raymond worked the puck behind the net before picking out Pettersson in front of the net, and he scored his second.
The win was sealed with just under nine minutes left, when Raymond showed wonderful skill on the left wing to outfox his defender and coolly tuck the puck home.
Raymond was then guilty of slashing, which allowed Dalibor Dvorsky to score on the powerplay in the last minute and add what was a consolation in terms of the result, but renewed Slovak hopes of sneaking into the automatic Quarter-final places, despite a 5-3 victory for Sweden.
The Swedish win means both they and Slovakia finish the group with six points. Sweden can only top the group if Italy takes at least a point against Finland later today.
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