Southampton moved out of the bottom three despite a 96th-minute Tom Davies goal that denied them victory at Everton.
The Saints go above Swansea, who they visit in a crucial game on Tuesday, on goal difference after Nathan Redmond’s second-half goal gave them the lead.
Winger Redmond nodded Cedric Soares’ cross past Everton goalkeeper Jordan Pickford in an otherwise subdued contest which saw Southampton’s Maya Yoshida sent off for two fouls.
Everton, who stay eighth in the table, were booed at the final whistle of their final home game of the season, and the result means they can no longer overhaul seventh-placed Burnley who have qualified for the Europa League.
Pickford denied the visitors twice in the first half, saving Charlie Austin’s sweetly-struck 20-yard volley and Oriol Romeu’s low deflected strike.
Even after Yoshida’s dismissal Southampton continued to threaten with Redmond’s clever running causing problems for the hosts.
But it took a vital save by Alex McCarthy to turn away Leighton Baines’ goal-bound free-kick before Davies’ strike deflected in off Wesley Hoedt in added time.