The Hammers remain on 30 points, just six above the bottom three with a tough away match at Stoke this weekend.
"We need a win but I don't think we're getting twitchy," O'Neil said.
"We're conscious of the fact we need to get to 40 points but up to now, every time the gap between us and the clubs below closes a bit we manage to produce the result that we need.
"I expect that to happen again and believe we can stay up comfortably. The quicker we get it done, however, the more we can relax for the rest of the season."