Saints climb as high as third, at least until Leicester play later on Sunday, and are only a point behind top two Tottenham and Liverpool, who are also both still to play.
The season is only 12 games old, but Sheffield United’s situation is looking increasingly dire. They still have only one point and will be at least six points from safety regardless of Sunday’s other results.
No team in the Premier League era has had so few points at this stage of the season and gone on to avoid relegation.
In such a lofty position with almost a third of the season completed, and now with only one defeat in their past 10 games, it is right to consider what this accomplished Southampton team could achieve this term.
Their defence is organised, their midfield a combination of hard-working and creative, while front two Adams and Danny Ings are among the most potent attacking forces in the league.
Bar a brief period of Blades pressure at the beginning of the second half, when imposing centre-halves Jan Bednarek and Jannik Vertergaard were made to work hard, Saints controlled this game.
Oriel Romeo patrolled in front of the defence, James Ward-Prowse constantly teased with his delivery from set-pieces and Adams was a nuisance to the visiting backline.
Given their superiority, particularly in the first half, Southampton perhaps could have won by more, and it is true they had a helping hand for their first two goals.
The first came from some weak Sheffield United defending of a Ward-Prowse free-kick. Bednarek leapt with barely a challenge and, when the ball looped off his shoulder, former Blade Adams nipped in to hook past Aaron Ramsdale.
After riding the wave of United pressure, Saints swept forward for the second. Ings, making his first start since the beginning of November after a knee injury, cut in from the left to feed Armstrong, whose shot took a wicked deflection to wrong-foot Ramsdale.
With the lead doubled, relaxation came over the 2,000 fans returning to St Mary’s, and Redmond’s sweet curler after a one-two with Romeo was no more than Saints deserved.