The win left Chelsea in 11th position, a point and one place behind Newcastle but with a game in hand.
Chelsea eased some of the pressure on coach Mauricio Pochettino with a 3-2 home win over fellow mid-table side Newcastle United in the Premier League on Monday with Cole Palmer, the Blues' best player this season, creating one goal and scoring another.
Chelsea's Senegal striker Nicolas Jackson diverted a Palmer shot past goalkeeper Martin Dubravka in the sixth minute before the hosts allowed Newcastle to get an equaliser in the 43rd when Alexander Isak curled a shot beyond the diving Djordje Petrovic.
Chelsea, urged on by their fans after some had turned on Pochettino and the club's US owners in a 2-2 draw at Brentford nine days ago, restored their lead in the 57th when Palmer, the Blues' top scorer this season who is in contention to go to the Euros with England this year, scored from outside the box.
Substitute Mykhailo Mudryk made it 3-1 in the 76th minute when the Ukrainian danced through Newcastle's defence, rounded Dubravka and slotted home. A fierce shot by Jacob Murphy gave the visitors hope in the 90th minute but Chelsea held on.
The win left Chelsea in 11th position, a point and one place behind Newcastle but with a game in hand. Another three points would push the Blues up to eighth and within sight of qualifying for the Europa League or the Europa League Conference.
Both clubs have failed to live up to the hopes of their deep-pocketed owners with Newcastle manager Eddie Howe under just as fierce a spotlight as Pochettino at Chelsea.
It was the Argentine who could breath a sigh of relief on Monday as the home fans left Stamford Bridge singing the Bob Marley song chorus "Don't worry about a thing", a sign perhaps that most of them remain patient with Pochettino's attempts to build a team from his expensive but young squad.
"We played a very good team with amazing players, it was really tough," he told the BBC. "We made the victory, but we keep looking forward and moving forward - the most important is the three points.
"We are building something different and we need to accept it. Today the fans were good, we need to keep moving and being strong."
Chelsea's only hope of silverware now lies in the FA Cup and they have a last-eight tie against second-tier Leicester City at home on Sunday. Newcastle must try to bounce back in time for their own quarter-final - a trip to Manchester City on Saturday.
Howe said he was frustrated that Newcastle were undone by sloppy defending.
"I thought we were good in lots of aspects of our play tonight," he told talkSPORT radio. "You can't concede goals like that, home or away, and expect to win the game, simple as that."
