Two goals from John Salako and one apiece for Marcus Hall and Simon Haworth gave Gordon Strachan's Sky Blues their best strike return since Blackburn were hit for five in December, 1995.
And it handed the Goodison manager, Howard Kendall, the massive task of lifting his underachievers, who were dumped out of this third-round tie and left to tread water in the Premiership's bottom three.
Indeed, Kendall's frustration boiled over after the final whistle, when he seemed to have stern words with Craig Short as Everton's players trooped off without acknowledging their travelling fans.
Coventry were without the suspended Dion Dublin and the injured trio of Noel Whelan, Darren Huckerby and Paul Telfer, but rookie strikers Haworth and Kyle Lightbourne never gave the visitors a moments peace.
Barmby did lift Everton's hopes with a 16th-minute opener after Hall had fired home his first goal in 61 games for the Sky Blues in glorious fashion after just five minutes, but that was his only telling contribution.
Coventry went ahead again on 33 minutes when Salako walked in his second goal of the season after Paul Gerrard failed to hold Lightbourne's drive.
Everton stepped up a gear after the restart, but their reliance on trying to find the out-of-sorts Ferguson with hopeful long balls was too predictable.
Two goals in two minutes settled the issue. First Salako scampered away down the right and cut inside the struggling Dave Watson before beating Gerrard with a perfectly measured chip. Then Haworth drove in his first goal for the club from a tight angle.
It could have been even worse for Everton since Lightbourne clipped a shot against the post and Hall rammed his follow-up against the bar.
Coventry City: Ogrizovic; Shaw, Burrows, Williams, Breen, McAllister, Salako (Boland 84), Lightbourne, Hall, Haworth, Nilsson. Substitutes not used: Soltvedt, Ducros.
Everton: Gerrard; Short, Hinchcliffe, Bilic, Stuart, Williamson, Watson (Ball, 71), Barmby, Ferguson (Cadamarteri, 71), Oster, Speed. Substitutes not used: Barrett.
Referee: S W Dunn (Bristol).Reuse content