"We couldn't have put it in our hands (against Wigan) but we could have put severe pressure on Everton and we failed to do that," O'Neill said.
"But I wouldn't be giving up. I think, looking at Everton's matches, winning Premier League games remains as difficult this week as it was last week and the week before and the last couple of years before that."
Even if Everton do manage to edge out the midlands club, O'Neill is upbeat about the possibility of his side's involvement in the frequently maligned Intertoto Cup.
"We have put our names down for the Intertoto Cup and if we finish sixth we would be guaranteed that," said the former Celtic boss. "If we win against West Ham, which is a tall order, we can't be overhauled for sixth.
"The Intertoto is not as stringent or severe as it used to be, where it was a case of getting ready in June for games. It's now one main game at a time when you would be playing pre-season anyway."