Arsenal remained in fourth place after Gabriel's header secured a 2-1 win at West Ham to stay two points ahead of Tottenham after its north London rival beat Leicester 3-1 on Sunday through Son Heung-min's double.
Chelsea looks well placed to qualify for the Champions League despite now being only three points ahead of Arsenal with four games remaining after losing 1-0 at Everton.
It's a result – secured by Richarlison – that gives Everton renewed hope of avoiding relegation after a 68-year stay in the top division.
Everton moved two points from safety with five matches to go and a game in hand over both Leeds and Burnley, which are just above the drop zone.
Richarlison’s four goals in his last five league matches is as many as he scored in his first 21 games of the season.
The latest came 59 seconds into the second half at Goodison Park. Richarlison and Demarai Gray pressured Chelsea defender Cesar Azpilicueta into losing possession and the Brazilian benefited by sweeping a shot past goalkeeper Edouard Mendy.
But Richarlison could face Football Association action after picking up a lit blue flare during his goal celebrations and throwing it off the field, although it was unclear where it landed.
Everton relied on Jordan Pickford’s saves to preserve the win – denying Cesar Azpilicueta, Ruben Loftus-Cheek and Thiago Silva.
"We’re still in a massive scrap," captain Seamus Coleman said. "We’ve got to keep going and keep battling."
Third-place Chelsea has only four points from its last four games.
"We struggle to play without big mistakes," Chelsea manager Thomas Tuchel said. "And that is why we struggle to have results."
Son is now on a career-high 19 Premier League goals in a season, only three behind Liverpool striker Mohamed Salah in the race for the Golden Boot.
Before scoring against Leicester, Son also provided the opener in the 22nd minute by delivering a corner for the unmarked Harry Kane to head into the bottom corner to end a five-game goalless run.
It also was Tottenham’s first attempt on target in 221 minutes.
The host struggled for another sight of goal until the hour mark when it doubled its lead.
Cristian Romero was the architect as he produced a crunching tackle on Çağlar Söyüncü on the halfway line which set Dejan Kulusevski free and the Swede raced clear and teed up Son, who found the bottom corner.
Son’s second came after he picked up the ball on the right, cut inside and curled a shot high into the net in the 79th minute from distance. The South Korea forward was denied a shot at a hat trick after being substituted by manager Antonio Conte
"We are talking about a great player, fantastic player," Conte said. "But it’s important at the end of the season to celebrate a big achievement with the team. Otherwise it remains only a personal success to score these types of goals."
Kelechi Iheanacho scored a stoppage-time consolation for Leicester, which is 11th in the standings and focused on its Europa Conference League semifinal against Roma on Thursday after drawing 1-1 in the first leg.
Like Leicester, seventh-place West Ham's focus is on Europe and its second-leg game in the Europa League semifinals against Eintracht Frankfurt on Thursday.
Arsenal still found it hard going before Bukayo Saka's corner was headed in by Rob Holding in the 38th minute for his first-ever Premier League goal since making his debut in 2016.
The lead lasted seven minutes as Jarrod Bowen leveled from close range before the break. But Arsenal went in front again nine minutes into the second half when Gabriel Martinelli crossed to an unmarked Gabriel for a diving header that beat former Arsenal goalkeeper Lukasz Fabianski at the far post.
After three straight losses, Arsenal has now won three in a row.
"It’s nice and close and we’ll have to keep going for the rest of the season," Holding said. "Your heart rate will be going high at times, but if we can keep controlling what we do and picking up points that’s all we can worry about."