This run of form has seen them climb out of the dropzone and Cork does not believe they will be sucked back into it.
"They should do," Cork told WalesOnline when asked whether Swansea will stay up this term.
"It's the same thing as last season, a little bounce when the new manager comes in. The mentality changes and you can see that on the pitch now.
"The club has got a bit of momentum back again and I hope they stay up. I've got a lot of friends here and it [relegation] would obviously effect them and other people at the club. It would be great if they could stay up."
The victory over Burnley fired the Swans up to 16th place in the table, one point clear of the relegation zone.