The tale should be about how a side who came into this trip to a top-six Premier League contender on the back of seven successive away defeats more than matched their illustrious hosts for ninety minutes, with Carlo Ancelotti - a man who knows football - entitled to be mightily relieved that his Blues side secured a fortuitous extra half hour, let alone progressed.
The hosts regathered their composure with substitute Abdoulate Doucoure booking the Toffees’ place in round four, but it was what went on before which will have concerned Ancelotti.
It started well enough for Everton, courtesy of Cenk Tosun’s opener. But that was as good at it got in regulation time.
After a rough start, Rotherham’s rally was outstanding. It displayed heart, character, commitment, belief and no little quality and showed the very best of the Millers under Paul Warne.
A headline moment saw Matt Olosunde level with his first-ever senior goal early in the second half.
The Millers did more than enough to progress in normal time, but Everton would ultimately prevail. Not without a hell of a scare.
The bottom line is if the Millers show similar assertion and conviction as they return to league business, they have every chance of achieving their aims of survival. It was a performance to be proud of, even if the opening was hazardous.
The evidence of the opening salvos of this game suggested a mismatch between sides separated by 36 clubs. It looked like the gap should have been much wider. But it would change.
The Millers’ anxious start when they did not look convincing at all was compounded by an opener that was smart and eye-catching from an Everton perspective, but less so from the visitors.
Michael Ihiekwe, back in his home city, got too tight to young midfielder Anthony Gordon, who hared away in a flash with his cutting pass finding the untracked Tosun, who delicately chipped the ball past the advancing Jamal Blackman for his first goal for the Blues since November 2019.
Tosun, whose future at Goodison is uncertain, was almost in again soon after another defence-splitting pass from Tom Davies and it had the makings of a long afternoon for the visitors.
Creditably, the response was forceful and much more in character as they started to be much more assertive and take the game to the hosts and turn it into a genuine Cup tie.
Everton - who made eight chances from their last fixture with 14-goal Dominic Calvert-Lewin and Jordan Pickford not involved at all - were pushed back a fair bit as the Millers started to compete and win battles and look dangerous from set-pieces.
The first sign of life saw Michael Smith fire over into the Gwladys Street End after a neat move involving the recalled Matt Crooks - who asserted himself on proceedings - and his reaction and that of the Millers’ bench suggested he should have done better.
A frenetic spell of action then saw the visitors go close on three occasions.
Olosunde cut inside before seeing his low shot blocked by the legs of Robin Olsen and after a blue shirt got in the way of Crooks’ follow-up, Dan Barlaser seized on the loose ball and his drive was turned away by the Everton custodian.
It was much better from the Millers as they set about the Toffees with an aggressive press and they went close again when Crooks headed onto the roof of the net after an excellent inswinging corner on the left from Barlaser.
Ihiekwe soon after profited from Smith’s response, with Everton’s only response before the break coming when Tosun steered wide after some nice inter-play between Seamus Coleman and Alex Iwobi.
After a strong spell in the second half of the first period, the onus was on the Millers to sustain it against an Everton side who came out early ahead of the resumption.
The Millers continued to take the game to the hosts, whose chief threat appeared to arrive on the break.
A quick counter ended in Iwobi dragging a shot wide, while at the other end, United continued to push with Smith failing to tee up Crooks when well placed after a good move from the visitors.
The Millers persisted and got their deserved rewards on 56 minutes.
Smith’s deflected pass found the onrushing Olosunde, who slid the ball home to level it up and Everton could have few complaints.
Enthused, Rotherham continued to set about Ancelotti’s passive side and more enterprise from Olosunde found Wes Harding, who blazed over as Everton’s afternoon started to deteriorate somewhat.
Clearly perturbed, Ancelotti made a raft of changes in a bid to change the narrative of a game which was being increasingly dominated by the visitors - while no doubt being observant that some of his players were starting to go missing.
One or two got the message as the clock ticked down, but Ancelotti was entitled to be a worried man and no doubt seething inside at what he had seen with his side fortunate to be level, in truth.
One moment of danger saw the alert - and under-employed - Blackman deny Tosun, while at the other end, George Hirst almost turned in a cross from fellow substitute Clarke.
It then looks as if Tosun had provided Everton with ‘get-out-of-jail card’ after sliding home from Rodriguez’s expert free-kick, but VAR decreed that the ‘goal’ was marginally offside, with the Millers’ endeavour earning that reprieve.
Milers secured an extra half hour of extra time, which was the very least their efforts merited.
Doucoure’s incisive finish would enable Everton to regain control when he coolly slotted home from Rodriguez’s pass, but there was no cause for celebrations from Ancelotti. More a spot of relief.
Suddenly, Everton started to fancy it again with Bernard striking the post and Tosun firing wide and Coleman stinging the palms of Blackman.
The Millers, with big hearts, did not give in. On the evidence of this afternoon, they were never going to.
Everton: Olsen; Coleman, Godfrey, Keane, Digne (Mina 66); Davies (Sigurdsson 66), Gomes; Iwobi (Doucoure 61)i, Rodriguez (Nkounkou 95), Gordon (Bernard 61); Tosun. Substitutes unused: Lossl, Holgate, Branthwaite, Simms.
Rotherham United: Blackman; Olosunde (Jones 90), Wood, Ihiekwe, A MacDonald, Harding (Clarke 83); Lindsay (Jozefzoon 76), Barlaser, Wiles; Crooks (Vassell 67); Smith (Hirst 77). Substitutes unused: Johansson, Vickers, Tilt, Miller.
Referee: S Attwell (Warwickshire).