Huddersfield and Crystal Palace took full advantage with vital wins over Watford and Brighton to leave Southampton five points off safety.
Mark Hughes looked set for a first league win as Southampton manager as they led the reigning champions 2-0 through Dusan Tadic and debutant Jan Bednarek with just 20 minutes remaining.
However, Olivier Giroud came off the bench to finally kickstart his Chelsea career with his first two Premier League goals for the club either side of Eden Hazard's leveller.
Victory took fifth-placed Chelsea to within seven points of Liverpool and Tottenham Hotspur.
"As long as it is mathematically possible we will believe we can reach the top four," added Giroud. "We have got five finals to play, after that we will see."
Liverpool can re-open a 10-point gap later on Saturday should they see off Bournemouth at Anfield.
And Spurs can also secure their place in the top four by inflicting a fourth straight defeat in all competitions on runaway league leaders Manchester City at Wembley.
Crystal Palace had been set to drop into the bottom three on goal difference prior to Chelsea's comeback.
However, they now enjoy a six-point cushion over Southampton thanks to a thrilling 3-2 win over Brighton at Selhurst Park.
Wilfried Zaha scored twice for the hosts as all five goals arrived before half-time.
James Tomkins was also on target for Palace, whilst Glenn Murray and Jose Izquierdo's replies for Brighton failed to halt their run of five games without a win.
Huddersfield look well set for just a second ever season in the Premier League as Tom Ince netted in stoppage time to secure a first win in six games as a 1-0 victory over Watford opened up an seven-point lead on the bottom three.
Swansea couldn't make home advantage count against Everton, but also opened up a five-point lead on Southampton just above the relegation zone.
Everton were outplayed at the Liberty Stadium, but Swansea proved their own worst enemies as Kyle Naughton's own goal gifted the visitors a half-time lead.
Jordan Ayew salvaged the point the hosts at least deserved 19 minutes from time.
Winning 🙌 pic.twitter.com/RNMyTRe64L
At the other end of the table, Burnley closed to within two points of sixth-placed Arsenal as a fast start was enough to secure a 2-1 win over Leicester.
Chris Wood and Kevin Long put the hosts 2-0 up inside nine minutes, and Sean Dyche's men resisted a second-half Foxes fightback despite Jamie Vardy's 20th goal of the season cutting their lead in half in the 72nd minute.