Newcastle were dragged back in to the relegation mix after losing two straight matches to Arsenal and Crystal Palace, but this victory now sees Newcastle move 10 points clear of the drop zone.
Benitez's contract expires at the end of the season and he has compared talks with controversial club owner Mike Ashley to Brexit negotiations.
"You cannot go up against a team thinking "maybe we can win" or that a draw is a good result, no. It's easier with this commitment from players". "The team has worked really hard this season and we can say we are a Premier League team for next season". We have the city, we have all the potential. It's massive. I'm not looking to spend £200million, but doing the right things. In the first half we were too slow with the ball, whenever you play against that type of team that defend deep you have to be quicker and cleaner through the defensive lines and I don't think we did that quick enough.More news: Georgetown University students vote on fee for slavery reparations
Leicester remain seventh on 47 points, while Newcastle moved up to 13th on 38 with their first away win since December.
Miguel Almiron then forced Schmeichel into a low stop as the visitors pushed for a surprise second.More news: Protesters March in Tehran Against US Decision to Blacklist IRGC
Harvey Barnes was a constant threat for Leicester and he surged past two defenders for a shot which Dubravka was equal to.
It had been a textbook counter-attacking away performance from Benitez's side and they began the second half on the front foot - only for Perez to horribly miscue Rondon's cross.
His strike partner Salomon Rondon had earlier hit the bar with a curling free-kick while Jamie Vardy spurned Leicester's best chance in the 80th minute when he scooped over from an excellent position in the box.More news: The Newest: Fed rate cut easing is wanted by Trump
The last opportunity for a Leicester leveller came and went in stoppage-time when Tielemans appealed in vain for a penalty after going to ground when a cross reached him at the far post.