Despite being denied back-to-back wins for the first time this campaign with near enough the final kick of the game, Southampton still climb out of the Premier League relegation zone and sit above next opponents Swansea City on goal difference.
Everton made a couple of enforced changes for their final home match of the season due to knocks sustained by Theo Walcott and Wayne Rooney, the latter of whom requires a scan to determine the extent of the damage.
Just one alteration was made by the Saints, meanwhile, bringing in Pierre-Emile Hojbjerg for Redmond, and it was the Saints who just about edged things in the first half.
That aside, however, openings proved to be hard to come by at both ends of the field in a forgettable first half on Merseyside.
Home boss Sam Allardyce changed things at the midway point by bringing on Ramiro Funes Mori for Yannick Bolasie and altering formation, but it was the switch made by Mark Hughes that proved more profitable.
Southampton, now with one win to their name in 19 Premier League visits to this ground, were made to see out the final five minutes with 10 men after Maya Yoshida picked up two silly yellow cards.
Alex McCarthy was required to tip Leighton Baines's free kick over the bar in the first minute of added time, before Davies struck two minutes after the allotted period to cruelly deny Southampton, with the final touch coming off the unfortunate Wesley Hoedt.
The result ensures that West Bromwich Albion still have a chance of surviving, having come within seconds of being relegated to the Championship, while Burnley are now guaranteed seventh place and a spot in the Europa League qualifiers.
EVERTON (4-3-3): Pickford; Baines, Keane, Jagielka (Klaassen 80'), Coleman; Schneiderlin, Gueye, Davies; Bolasie (Funes Mori 46'), Tosun, Vlasic (Niasse 58')
SOUTHAMPTON (3-4-2-1): McCarthy; Bednarek, Yoshida, Hoedt; Cedric, Romeu, Hojbjerg, Bertrand; Tadic (Ward-Prowse 70'), Lemina (Redmond 46'); Austin (Long 77')