Heading into final Sunday when they host West Ham United, Leicester are 18th with 31 points, two less than Everton who are at home to Bournemouth on the final day and above 19th-placed Leeds on goal difference.
Leicester also have a better goal difference than Everton so if they beat West Ham, Everton will also have to beat Bournemouth to preserve their 69-year tenure in the top flight.
Smith, whose side defended for the vast majority of the game at Newcastle and were saved by the woodwork three times, said that at least they have still have a chance after many had written off the 2016 champions.
"I think we have (put pressure on Everton). If they draw and we win we stay up. I did say it might be down to goal difference," Smith, who took over until the end of the season following the sacking of Brendan Rodgers in April, said.
"We have got a tough game against West Ham who have just qualified for a European final. Their manager is a good friend of mine, David Moyes, he's someone I have a lot of respect for and he will make it a really tough game for us.
"But we've made Everton have to win if we win ourselves. That performance (against Newcastle) was what supporters deserved with all the travelling they've done this season, and it was a clean sheet they deserved."
It promises to be a nerve-jangling final day for fans of Leicester, Everton and Leeds.
"We're going to need help. I think there's going to be a lot of anxiety between the three teams that are in it," Smith said.
"If Everton win... we obviously will give it our best shot and as long as we can say that we have fought tooth and nail, all we can do is put pressure on them."
