The biggest night in Everton’s season started with a big call from David Moyes and demanded a big performance from his players. With the exception of Jack Grealish, they did not let him down. A quarter-final at Fleetwood Town is their prize after a clinical, committed display shattered Newcastle’s hopes of lifting the Carabao Cup for a second time in three seasons.

Newcastle dominated spells of the fourth round tie and the stats – registering 28 shots to Everton’s 11, although only nine visiting attempts were on target – but individual defensive errors and an ineffective attack proved a ruinous combination. Matthias Jaissle denied his team are a soft touch, despite persuasive evidence to the contrary, yet called Newcastle “naive” and the defeat, in front of almost 8,000 of their fans, “shocking”. Andy Burnham was among the many Evertonians in attendance grateful for Newcastle’s generosity.

James Garner, with his first goal of the season, Hayden Hackney, with his first goal for the club, and Thierno Barry were on target to take Everton into the last eight of a competition they have never won. Jordan Pickford was in commanding form against his old adversaries while Kiernan Dewsbury-Hall ran the show from midfield. But it was the absent Grealish who dominated the buildup after Everton announced he had been dropped for disciplinary reasons.

Grealish is understood to have missed training on Monday without permission. Despite the importance of the fixture and the limited options at his disposal, and the playmaker’s subsequent return to training, Moyes demonstrated that no player is bigger than the club by banishing the 31-year-old from the game. The Everton manager later claimed it was “a club decision” but it was taken by him and the chief executive, Angus Kinnear. Ultimately, and no doubt to Grealish’s great relief, he was not missed.

Newcastle dominated the opening 20 minutes to the extent that a two-goal interval lead for Everton would have seemed inconceivable. Pickford saved an early header from Harvey Barnes, who then turned provider to set up Jacob Ramsey for a shot that was blocked superbly by Vitalii Mykolenko. Fabian Schär also tested Pickford from distance, leaving the England international raging at the lack of protection in front of him. He could not fault his defenders’ willingness to put their bodies on the line, however.

Everton gradually turned the contest around. Putting an end to ineffective long balls up to Barry, when he was marked by Dan Burn, helped their cause, although it was a long clearance from Pickford that led to the breakthrough. Amar Dedic looked to retrieve the goalkeeper’s punt but was harried out of possession by Tyrique George. The ball broke to Hackney, who squared inside for Garner to find the far corner of Lukas Hornicek’s goal. “The first goal, it’s no risk, clear it,” said a dismayed Jaissle. “But we concede out of nothing. The first, second and third goals were the same with individual mistakes. We need to be more efficient and learn fast.”

Pickford came in for his customary Sunderland stick from the Newcastle support. He responded in the best way possible; with a hand in a second goal that deflated the sizeable away section. The Everton keeper reacted sharply to launch a quick throw towards Dewsbury-Hall in the centre circle. Dewsbury-Hall spun away from Burn before embarking on a determined run that was halted by a trip from behind by Ramsey. The referee, Farai Hallam, played an excellent advantage as the ball broke to Barry. Newcastle appeared to pause, waiting for the free-kick, and were punished when Barry clipped over a cross for Hackney to volley home from close range.

A triple half-time substitution from Jaissle was an unmistakable sign of discontent with Newcastle’s ineffective display at both ends of the pitch.

There was more menace from the visitors in the second half with the substitute Yoane Wissa having an early chance from Joe Willock’s chip and Dedic’s shot looping over off Hackney’s heel. But with Pickford standing firm and saving smartly from Lewis Hall, Anthony Elanga and Ramsey, and Wissa almost hitting the corner flag with one decent chance, Newcastle’s failure to make their pressure count continued.

The visitors were punished when Everton made the game safe in devastating style. Garner, a growing influence, intercepted a pass from Dedic and released Dewsbury-Hall behind a static Newcastle defence. The midfielder took Burn out of the equation with a perfect pass across to Barry, who had time to control and place a fine finish beyond Hornicek.

Burn finally broke Everton’s resistance with a late header from an Elanga cross. A melee involving the goalscorer, Pickford, James Tarkowski and Joelinton broke out as they fought over the ball – all four were booked – but there would be no miraculous escape for Newcastle. It is Everton who march on to face the Cod Army.