The 3-0 loss at Goodison Park, which saw the Magpies concede two late penalties and have substitute Jamaal Lascelles sent off in the dying seconds to cap a poor team display, left Steve McClaren's side two points from safety.
Saturday's visit of West Brom, who are eight points ahead in 14th, takes on huge significance and Elliot accepts the players owe supporters something.
"I can't speak highly enough of the fans. I can only apologise that we didn't put in the performance and get the result we should have got," said the goalkeeper, who was the only player to escape any blame after pulling off a string of saves.
"We will do everything we can to put it right for Saturday."
Newcastle have lost 15 of their last 18 games away from St James' Park, something Elliot knows cannot continue if they are to survive.
"We do need to look at our away form because it is something we need to address," he said.
"Performances prior to this haven't been as bad as results have suggested, but we need to find something to get us going away from home.
"We have to hold our hands up about that.
"I think we got what we deserved. Now we need to make sure we kick on for Saturday.
"It's one of those situations where no-one is pulling away down the bottom. Everyone is stalling at the first fence.
"It is important for us to get some results and get some points going because if we could get them now, we could pull clear of trouble.
"We need to keep more clean sheets, we need to score more goals, we need to do everything better.
"With the signings we've made and the progress we've made on the pitch, hopefully it's not far off coming."