Ashley Young’s miss in the penalty shoot-out put Southampton through in the Carabao Cup at Everton’s expense.
Young, who had earlier been booed when he came on as substitute, saw his spot-kick saved by Alex McCarthy after the first 11 kicks had all been converted.
In a tale of two captains, Saints’ Taylor Harwood-Bellis had earlier cancelled out Abdoulaye Doucoure’s opener for Everton and then there were jeers when Dyche hooked the team’s only striker Beto for Young midway through the second half.
Neither side on show had gained a league point this season and there were 18 changes made between them.
Everton were quicker to settle despite deploying Dwight McNeil at left-back and giving first starts for local 17-year-old Harrison Armstrong and Belgian international Orel Mangala.
Beto had already forced Alex McCarthy into a save at his near post when Sean Dyche’s side broke the deadlock after 20 minutes.
The Saints ‘keeper flapped at a McNeil corner and when the ball broke for Michael Keane he nodded back into the box where Doucoure netted with a glancing header.
Only Lesley Ogochukwu from the visitor’s line-up survived from the weekend defeat at home to Manchester United and Russell Martin’s side looked disjointed as Everton’s Jesper Lindstrom was left clear to bear down on goal before finishing weakly at McCarthy.
When Beto found himself in a similar position, he was outmuscled by skipper Harwood-Bellis and his fall to the floor looking for a penalty fell on deaf ears.
The squandering of chances cost Everton when the visitors equalised after half-an-hour. Charlie Taylor whipped in a free-kick where Harwood-Bellis was allowed by the home defence to rise above McNeil and head down past Joao Viriginia for his first goal since last December.
Saints sent on Mateus Fernandes at the interval but their 75% possession continued. Everton’s approach was more direct but after 54 minutes Lindstrom was again denied one-on-one by McCarthy and flopped on the turf in dismay. At the other end, Virginia did well to keep out Ryan Fraser at his near post.
Everton fans’ displeasure was aired when Dyche withdrew Beto for Young though it allowed McNeil to push forward to join Iliman Ndiaye in attack.
McNeil’s looping shot deflected over the bar as both sides chased a late winner to avoid penalties.
Saints appealed for a late spot-kick in the 89th minute when youngster Tyler Dibling appeared to be fouled by Young but referee Darren Bond didn’t give it.
The 39-year-old former England international’s luck ran out in the shoot-out however. With every kick converted, he stepped up and McCarthy dived to his right to put Saints into the next round.