James Tarkowski broke the deadlock by combining with Dwight McNeil to power home a looping header against their former club in the 13th minute.
The Blues doubled their lead in the second half when Dominic Calvert-Lewin knocked down McNeil’s corner for Amadou Onana to score from close-range.
Ashley Young added a third in stoppage time as produced a deft finish inside the penalty area to meet substitute Beto’s surging run and cross at the byline.
Sean Dyche’s side now face another home tie with Fulham in the last eight.
Here were the key talking points from Goodison Park:
Burnley undone by Dyche’s secret weapon
The reunion with Burnley was always likely to set the tone for this encounter.
Sean Dyche spent the best part of a decade at Turf Moor’s helm while Dwight McNeil and James Tarkowski’s respective careers took flight with the visitors.
Impartial observers could be forgiven for believing the trio departed in bad blood after McNeil was booed mercilessly by his former fans from the outset.
But the winger responded in kind as Everton opened the scoring early on.
McNeil and Tarkowski had previously combined to hand Dyche a flying start to his tenure with a goal made in Lancashire for last season’s win over Arsenal.
They also repeated the trick to give their side the lead in September’s long-awaited maiden Premier League victory of the season against Bournemouth.
A follow-up arrived after just 13 minutes on Wednesday, when McNeil sent in a teasing cross for his former Clarets’ cohort to meet with a looping header.
McNeil and Tarkowski’s axis is fast becoming Everton’s secret weapon and their former club were simply its latest victims beneath the Goodison lights.
Goodison strikes the right note
A poignant evening had been anticipated following Bill Kenwright’s passing.
Whatever the debates over the late chairman’s legacy, the death of a lifelong Evertonian deserved a fitting send-off before the club’s latest home game.
The Grand Old Lady struck the right note with a warm applause to celebrate Kenwright’s life as his family and Joe Royle lay floral tributes on the pitch.
Farhad Moshiri helped mark the occasion by ending a two-year hiatus from attending matches at Goodison to pay tribute to his late boardroom ally.
The majority shareholder’s last appearance here was for a dismal 5-2 loss to Watford in October 2021, some 35 home games and two managers ago.
Moshiri’s return to the directors’ box also marked a first public endorsement of Dyche on home soil by the club’s senior leadership since he took the reins.
Accompanied by prospective new owners 777 Partners, this rare visit may prove to be the British-Iranian billionaire’s last to the blue half of Merseyside.
Blues dare to dream of Wembley
Near the end of this comfortable win, the Gwladys Street set out its stall.
The terrace repertoire had followed a steady theme for most of the game, with a whole-hearted rendition of ‘Spirit of the Blues’ a second-half highlight.
But one stood out above all others as the semi-ritual chants of ‘We’re going to Wembley’ rang out minutes before Ashley Young racked up Everton’s third.
Showpiece appearances in the Carabao Cup have proved elusive in modern times, with Gordon Lee’s perennial nearly men of 1977 still sole achievers.
David Moyes and Roberto Martinez went closest by reaching the semi-final stage, agonisingly so in the latter’s case, yet it continued to carry limitations.
With Manchester United and Arsenal joining an already-deposed Manchester City in exiting the competition early, however, those chances have increased.
Another favourable home tie in the last-eight against Fulham on the run-up to Christmas enhances the prospect of Dyche’s side going one step beyond.
If they can overcome that challenge and earn another favourable draw in the next round, the Street End’s prophecy might finally come to fruition.
