Maurizio Sarri's side beat Watford 3-0 to lift them into third spot behind Liverpool and Manchester City and their day was made all the sweeter as United drew at relegated Huddersfield and Arsenal were held by Brighton.
The results mean that with one round of fixtures remaining Chelsea, on 71 points, cannot be caught by fifth-placed Arsenal, who are on 67 points, or United, a point further back.
Fourth-placed Tottenham, on 70 points and with a far better goal difference than north London rivals Arsenal, are virtually assured of Champions League football next season barring an unlikely sequence of events.
Sarri's team were booed off after a sloppy first half, but two goals in three minutes from Ruben Loftus-Cheek and David Luiz lifted the mood at Stamford Bridge before Gonzalo Higuain sealed victory, Chelsea's first in four games in all competitions.
Finishing in the top four and winning the Europa League would make the Italian manager's troubled first season a relative success.
"We want to be in the top four at the end of the season. We want to be in the Champions League," said Sarri, whose side drew the first leg of their Europa League semi-final against Eintracht Frankfurt 1-1.
"The Europa League is also a very important competition and we want to win it because we think we deserve to a trophy this season. So we have two targets."
Despite their miserable 1-1 draw, Arsenal can still reach the Champions League after two seasons away from Europe's top club competition if they win the Europa League.
They are in a strong position after beating Valencia 3-1 in the first leg of the semi-final.
Arsenal, who came into the match after three straight Premier League defeats, started brightly at the Emirates and took the lead through an early Pierre-Emerick Aubameyang penalty.
But they squandered opportunities to extend their advantage and were made to pay when Glenn Murray scored from the spot following the hour mark after Granit Xhaka fouled Solly March.
Arsenal poured forward in search of a winner but Aubameyang volleyed wide from close range and Brighton goalkeeper Mat Ryan made several fine saves to keep them at bay.
Disappointed boss Unai Emery turned his thoughts to Thursday's second leg of their Europa League tie, with Arsenal set to finish outside the top four for a third consecutive season.
"We knew it is going to be difficult but our focus is now the Europa League," he told the BBC. "We have the opportunity in the Europa League to do something important and we will try and do that."
- United embarrassed -
United slinked off the pitch at Huddersfield after an embarrassing 1-1 draw, with manager Ole Gunnar Solskjaer admitting his side did not deserve to qualify for the Champions League.
The visitors had to rely on Huddersfield for a helping hand for their only goal as Scott McTominay's strike from the edge of the box went straight through Jonas Lossl's to given them an early lead.
Huddersfield had only scored nine goals at home all season, but one hopeful punt from Lossl cut United open for the equaliser on the hour mark.
Luke Shaw failed to cut out the Danish goalkeeper's clearance and allowed Isaac Mbenza a clear run on goal to slot between David de Gea's legs.
United have now won just two of their past 11 games in all competitions, leaving the decision to hand Solskjaer the job on a permanent basis in March open to question.
"We gave ourselves a chance to be in the Champions League," Solskjaer said of a run of 14 wins in his first 17 games in charge before the rot of the past few months set in.
"We got so many opportunities to grab third or fourth and weren't able to. The Europa League is the right place to be for us next year."
Barring an astonishing Spurs collapse, the only issue that remains to be decided in the Premier League is the destination of the title.
Manchester City, two points behind leaders Liverpool but with a game in hand, host Leicester on Monday, with the final round of fixtures taking place next Sunday.