The goalless draw at Swansea at the weekend took Southampton to 39 points - eight clear of the drop zone - with four matches remaining in their first season back in the Premier League.
Cork told the Daily Echo: "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.
"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 made his 100th appearance for Southampton in the game at the Liberty Stadium and was happy to leave with a point.
"We had to deal with a lot of pressure from Swansea in the second half and at the beginning of the first half," he added.
"It was a hard game for us. The heat didn't help, especially when you're playing a team that likes to pass it round a bit, because we were going to have long spells without the ball.
"It was hard, but we also played some good stuff at times. We created a few chances and it was a good game. I think both teams would take the point."