Liverpool - Southampton's hopes of avoiding relegation took a blow as Everton midfielder Tom Davies scored an equaliser in the sixth minute of added time to rescue a 1-1 draw on Saturday.
Substitute Nathan Redmond's first goal in almost a year appeared to put the fight very much back in Saints' hands as it would have left them needing a win at fellow strugglers Swansea on Tuesday to guarantee safety.
However, Davies' deflected shot off defender Wesley Hoedt moved Mark Hughes' side above 18th-placed Swansea on goal difference only.
It was cruel end for the visitors, who finished the match with 10 men after Maya Yoshida's 85th-minute red card.
Southampton received a pre-match boost with the absence from Everton of Wayne Rooney (calf) and Theo Walcott (knee), and despite early half-chances from Davies and Cenk Tosun the visitors remained composed and controlled.
Twice, Everton goalkeeper Jordan Pickford was forced into action, saving Charlie Austin's half-volley and Oriol Romeu's shot which deflected off Phil Jagielka.
Southampton broke the deadlock 11 minutes into the second half when Redmond directed a far-post header between Pickford's legs from Cedric Soares' cross, his first goal since May 13.
Redmond's introduction deserved to be game-changing, although whether Saints boss Hughes would have made the change so early had Mario Lemina not been injured is open to debate.
The former England Under-21 forward had chances to add to his tally, notably tricking his way into the area to force another save from Pickford.
Yoshida's departure to a second yellow card put Southampton under unnecessary pressure and their resolve cracked deep into added time with Davies' deflected shot.