Andy Carroll believes West Ham are safe after their 2-1 win at Sunderland on Monday night.
The striker scored the opening goal at the Stadium of Light and a second from Mo Diame put the Hammers clear.
Sunderland pulled one back through Adam Johnson but it was not enough to earn them anything.
West Ham now have 37 points, are comfortable in mid-table, and Carroll echoed co-chairman David Gold's Twitter claim that they are out of trouble.
"You take every game as it is, keep an eye on the teams winning below, but I think we are safe, yeah," said Carroll.
Carroll's man-of-the-match performance again roused the debate as to whether or not he should be picked for England’s World Cup squad.
"That's what I'm looking for now, scoring goals and keeping on it until the end of the season," he added.
Hammers manager Sam Allardyce, who was booed by his own club's fans after last week's 2-1 victory over 10-man Hull, was a satisfied man as he departed the English north-east.
He said: "We have won six of the last nine matches, we are probably in the top four or five in the Premier League for results at the moment.
"It got us out of trouble and it's got us almost safe – not quite mathematically, but I think it would be difficult for anybody to catch us now."
By contrast, opposite number Gus Poyet cut a disconsolate figure as a must-win game ended in a defeat which left his side four points adrift of safety with eight games to play.
However, he remained defiant as he considered the task ahead.
He said: "Look, if my team gives up, I will leave. Nobody will need to sack me, I will leave on my own, so if you see this team giving up before the end of the season, you won't see me here."
Sunderland have now won only one of their last nine home league games and taken just a single point from the last 18 they have contested.
Asked if the Black Cats could stay up, Poyet said: "Well, if we play like we played against Liverpool and today in the next eight games, we are going to have a great chance.
"Now the crystal ball, I can tell you, is not in my house, so I am not going to try to read the future because I'm not very good at that."