Richarlison volleyed a 75th-minute winner to end the Toffees eight-match run without a Premier League away win and earn Silva a much-needed reprieve.
Hasenhuttl’s men remain entrenched in the relegation zone and were jeered off having suffering a fourth successive league defeat on home soil for the first time since moving to this stadium from the Dell in 2001.
Southampton were back in front of their own fans for the first time following their record-breaking 9-0 humiliation at the hands of high-flying Leicester.
Hasenhuttl admitted pre-match that this was a “six-point” game and used his programme notes to call for togetherness.
Under gloomy skies on a soggy south-coast afternoon, enthusiasm in the stands – which was aided by the distribution of free scarves – dissipated within four minutes as the visitors made a dream start.
Gylfi Sigurdsson’s outswinging corner from the right was nodded on by Mason Holgate, leaving the unmarked Davies with a simple headed finish at the far post to claim his first goal since scoring against Saints at Goodison Park 553 days ago in May 2018.
Dominant Everton were eager to reinforce their advantage with a second and would have done so but for recalled forward Cenk Tosun wastefully heading over at the end of an unconventional one-two with Theo Walcott.
Saints, without captain Pierre-Emile Hojbjerg because of a concussion suffered in training, were struggling to trouble the Toffees and failed to muster an effort at Jordan Pickford’s goal before being understandably booed off at the break.
Home supporters responded with self-deprecating chants of ‘we’ve had a shot’ when the Moroccan curled straight at Pickford moments after the restart, but they had genuine reason to cheer shortly after.
Boufal tricked his way past former Saints midfielder Morgan Schneiderlin on the right before his low centre deflected off Yerry Mina to give Ings, who had looked increasingly isolated in the first half, a simple 50th-minute tap-in.
A game which had been so one-sided in the opening 45 minutes was now back in the balance.
Everton midfielder Sigurdsson – playing in place of Andre Gomes, who suffered a horrific fracture dislocation to his right ankle against Tottenham last weekend – forced Alex McCarthy to tip over, while a James Ward-Prowse free-kick was repelled by Pickford at the other end.
Everton then stepped up a gear and went back in front 15 minutes from time.
Right-back Djibril Sidibe found space on the right and his inviting cross was thumped home by the Brazilian forward at the far post.
Southampton had run out of steam and could easily have lost by more than one goal.