Their first goal from Danny Graham - his first in his 28-month spell at the club - had a huge slice of fortune as the striker deflected Jordi Gomez's shot, and when Toffees midfielder James McCarthy's shot beat goalkeeper Costel Pantilimon but rebounded off the post there was a feeling it was to be their day.
Jermain Defoe then diverted home Adam Johnson's deflected shot for another messy goal - but what was important was a vital win which lifted the side to 36 points, with fellow strugglers Leicester to visit the Stadium of Light next weekend.
"We were lucky today. At some moments they outplayed us but our back five including the goalkeeper did really well," said the Dutchman.
"Some other players played well last week but they had more problems with Everton, but at the end of the day we got three points and I try to forget the nature of things.
"It was not a great game by our side but at the other end it was very effective and that is what counts.
"It is not only luck, they worked very hard. The defence stood behind us and if you have the commitment to give everything even against a better team, you can get results."