Brynäs was the hottest from the start and had a lot of sharp situations, not least in an early power play.
But Oskarshamn, with goalkeeper Tim Juel, persevered and were instead able to score the first goal of the match themselves.
Antti Suomela made it 1-0 after just over six minutes.
It was his 29th goal of the season, which is the most in the SHL.
The Finn extended his fine goal haul in the second period when he flicked in 2-0 from a really tight angle.
3-0 and 4-0 also came, but surprisingly not from Suomela's club.
However, Suomela scored his third goal of the game, 31 of the season, when he made it 5-1 in the third period.
In addition to Suomela, goalkeeper Tim Juel was the big match winner after saving a total of 51 shots.
Brynäs, with four straight losses, is now in free fall and is now in a negative qualifying position.
HV71 steps over
Parallel in Jönköping, HV71 had 2-0 against Färjestad after the first period, where Andy Miele accounted for both goals.
Färjestad reduced early in the second period, but HV71 pulled away to 4-1 before the middle act was over.
Färjestad chased feverishly to catch up with HV and also scored 4-2, via Joakim Nygård.
But the reigning Swedish champions didn't get any closer than that, which meant their third straight loss.
On the other side, HV71 were able to step past Brynäs and onto safe ground (for the first time since September) after winning four of their last five matches.