The result at a raucous Goodison Park means Frank Lampard’s side are just two points from safety with five games to play – and will leave Leeds and Burnley sweating.
After a goalless first half in which Everton had the better of the play, the hosts made the crucial breakthrough just after the restart.
Richarlison pressured Cesar Azpilicueta into losing possession and the Brazilian forward finished coolly from Demarai Gray’s pass in the 46th minute.
Just before the hour mark Everton had a huge let-off when a strike from England midfielder Mason Mount hit both posts before man-of-the-match Jordan Pickford sprinted across his line to keep out Azpilicueta’s follow-up effort.
The England goalkeeper also denied Antonio Rudiger at point-blank range and kept out a Ruben Loftus-Cheek shot from distance.
The home crowd faced the agony of seven minutes of added time but Everton held out against their third-placed opponents, prompting a huge roar of relief at the final whistle.
Former Chelsea boss Lampard, who took over from the unpopular Rafael Benitez at the end of January, has struggled to revive the fortunes of the Merseyside giants, who last played outside the top-flight in the 1953-54 season.
But he was delighted with the passion shown by his players and the supporters.
“The fans are in the middle of this,” he said. “I am pleased the players could see on the way in what it means to the fans.”
Son Heung-min scored twice as Tottenham eased past a weakened Leicester side 3-1 to climb to fourth, one point clear of Arsenal, who travel to face West Ham in the later kick-off.
Antonio Conte’s side had recently lost momentum in the race to qualify for the Champions League, taking just one point from their previous two matches with no shots on target. (AFP)