Manager Steve Bruce insists Newcastle must finish their Premier League survival job after Callum Wilson claimed a second-half brace in a stunning 4-2 win at Leicester.
Wilson's double added to first-half goals from Joe Willock and Paul Dummett as the Magpies shocked the Foxes.
Bruce's side will be safe if Fulham drop points against Burnley on Monday as the Cottagers can only match their tally of 39 points.
Marc Albrighton and Kelechi Iheanacho scored two late consolations for the Foxes, who suffered a potentially damaging blow to their Champions League qualification hopes.
Newcastle climbed to 13th on the table but Bruce promised not to relax in their survival fight.
"We are a darn sight better off than a month ago," he said. "We have made huge strides but we cannot take anything for granted.
"It's a nice easy one against Manchester City next week, until you get to the magic figure you don't know what it is but we have given ourselves a chance of getting past 44 points which is what we got last year.
"We are not safe, we are almost there. Until it is done we have to get ready for the next challenge."
Kasper Schmeichel had already saved brilliantly from Allan Saint-Maximin before Willock opened the scoring after 22 minutes, cashing in on Caglar Soyuncu's error.
Just 12 minutes later Dummett made it 2-0 when he headed in Matt Ritchie's corner for his first goal since January 2016.
Martin Dubravka made two smart saves from Wesley Fofana and Iheanacho, while Wilfred Ndidi headed wide but Wilson's double ended any hopes of a comeback.
He made it 3-0 when Timothy Castagne failed to cut out Ritchie's deep ball after 64 minutes and grabbed a second 17 minutes from time, tapping in after his initial shot came back off the post.
Albrighton's screamer and Iheanacho's low shot did pull two goals back but Leicester never recovered from losing Jonny Evans to his long-standing heel injury in the warm-up.
They remain third but have given Chelsea and fifth-placed West Ham - five points behind with a game in hand - a chance of catching them in worrying echoes of last season when the Foxes missed out on Champions League qualification on the last day.
"They have more resilience (from last season)," said boss Brendan Rodgers. "I have always said we have to fight for everything. We are so close to where we want to be.
"We are in a great position still. I have always said 'let's see where we are after 38 games'.
"We could have had four or five goals but if you defend like that you are going to struggle to win games.
"The blow of Jonny not playing (was difficult) because he is such a great organiser for us."