The Saints are currently 12th in the league table, eight points above the drop zone with just four matches left to play.
However, despite the fact that they are all but safe, Cork is eager for Southampton to end the campaign strongly.
"We've got West Brom at home and if we can win that then hopefully that will be us safe I think, but at the minute we just need a few more points," he is quoted as saying by the Daily Echo.
"We worked so hard last year and I was so happy for the lads to get up. For us to stay in the Premier League this year, I know how much it would mean to all of them."
Cork was speaking after his side had played out a 0-0 draw with Swansea City on Saturday afternoon.