Newcastle moved out of the bottom three for the first time since 1 February, as rivals Sunderland and Norwich slipped up in the fight for survival.
The 1-0 win over Crystal Palace saw them rise to 17th, and they have now picked up eight points from 12.
“We must keep the momentum,” said manager Rafael Benitez, with a trip to relegated Aston Villa next week.
Both the Black Cats and the Canaries have a game in hand on Newcastle.
‘We have to keep going’
Could the appointment of Benitez prove to be owner Mike Ashley’s greatest signing since he took control of Newcastle in 2007?
With £100m the reward for Premier League status next season, Ashley took the plunge and sacked Steve McClaren in March with the team in 19th position.
It took four matches for the former Liverpool and Real Madrid boss to get his first win, but now it is no defeats in the past four.
Andros Townsend scored a brilliant winner against Palace on Saturday, but Benitez also had keeper Karl Darlow to thank for saving Yohan Cabaye’s second-half penalty.
“The win could be important for us,” said Benitez. “I’m sure it’s important for the fans and everyone now when you see the table. But we still have to keep going.
“We know that we have to approach the next game against Aston Villa like another final, and that’s it. We cannot look too much at the table – we just have to be sure that the next game, we will be ready.”
‘We need to hold our nerve’
Sam Allardyce’s Sunderland side required Jermain Defoe’s stoppage-time penalty to rescue a point at Stoke.
The Wearsiders have now drawn their last two games and have only won one match in the past nine.
Commenting on Newcastle’s win, Allardyce said: “Those things are not in our control.
“What’s in our control is making sure that we try and win next week [against Chelsea] and then make the game in hand really count against Everton.
“That’s our big task in the next two games. If we can hold our nerve, we have the capabilities to see if we can try and do the escape that this club has done for the last two or three years.”
‘Thinking about the others will drive us crazy’
Alex Neil’s Norwich have stuttered in recent weeks.
They have now lost their past three Premier League games, the latest loss coming at Arsenal where the Gunners’ Danny Welbeck scored the only goal as a second-half substitute.
Having hovered just above the relegation zone over the past four weeks, they are now down to 19th.
“The last thing you can do down in the dogfight is feel sorry for yourselves,” said the Canaries’ Scottish manager.
“We didn’t get what we deserved at Arsenal. We need three displays like that in the next three games. If you start worrying about the maths and what other people are doing then it’ll drive you crazy.”
He added: “Newcastle are a different kettle of fish because they’ve only two games left but I think it’s unlikely Sunderland will win their three remaining games.
“We just have to try and take as many points as we can.”