Arsenal's European hopes suffered a self-inflicted blow in a 1-1 Premier League draw with Burnley at Turf Moor.
The Gunners were dominant and led through Pierre-Emerick Aubameyang's sixth-minute opener but contrived to gift the Clarets an equaliser in the 38th minute.
Arsenal were awarded a late penalty for handball when Nicolas Pepe's volley hit Erik Pieters and struck the crossbar.
The Burnley defender was also shown a red card but both decisions were overturned after replays showed the ball struck Pieters' shoulder.
A first win in 10 matches stopped Saints' alarming slide and moved them 10 points clear of the bottom three, but the rock-bottom Blades look more unlikely by the week to pull off the greatest of escapes.
The defeat leaves Chris Wilder's side 12 points adrift of safety with just 10 matches remaining to save themselves.
Ward-Prowse netted from the penalty spot in the 32nd minute after Nathan Tella was brought down and Adams' superb strike four minutes after half-time gave Saints breathing space.
The defender fired over an open goal from a yard out after Conor Coady hit the post.
Emi Martinez also denied Coady in the second half at Villa Park as the hosts drew for just the fourth time this season.
Ollie Watkins and Ezri Konsa hit the woodwork for Villa during a dominant first half but they lost their way after the break and relied on Saiss' howler to claim a point.