Sweden took the lead in the sixth minute as Erik Karlsson skipped past a defender and slotted home.
Only five minutes later, Alexander Popov equalised for the Russian team from close range.
But Henrik Zetterberg put Sweden ahead again, grabbing the rebound off Daniel Alfredsson's shot to score 14 minutes into the period.
Both Sweden and Russia have won all four of their games leading up to the match. Sweden's lead in the group was down to the team's superior goal difference.