In an incredible game at Trelleborg Rugby Arena, the visitors struck first before fashioning a 20-12 half-time lead.
Sweden cut the gap with a converted try before their opponents crossed the whitewash.
But another seven-pointer six minutes from time nudged the hosts in front.
And their fifth try of the afternoon a minute from time made it 33-25.
Denmark grabbed an injury-time consolation, but it was not enough.
Sweden remain third in the table with 14 points ahead of their final game in Poland on Sunday.
Their rivals are second with 16, both teams having won three out of four matches.
Unbeaten Czechia, the only team to beat both, are favourites to take the title on 18 points.
They can wrap the silverware up in Denmark on Saturday, but any slip-up would leave the door open for Axel and Co.
