Defeat to the Scandinavians leaves Serbia second from bottom in Group I with one point, having been awarded a 3-0 win for their abandoned fixture - though the points were docked for their involvement.
As part of their UEFA punishment, Serbia were forced to play Friday's clash in front of an empty stadium in Belgrade.
With Denmark and Portugal six and five points better off respectively in the pool, Serbia's chances of reaching the finals in France look in grave danger.
And Advocaat has walked away just five months after taking the job, apologising for his inability to spark a strong start to their campaign.
He said in a statement: "I am very sorry that my spell worked out this way.
"I take responsibility for everything I've done at the professional level in the period.
"The pressure we were under was clearly visible and was also too big for this team.
"I was confronted with a problem that I could not solve. I think I made the honourable decision for me, it was very difficult.
"I wish the Serbian team all the best for the future."