The most optimistic Evertonian – if that is not an oxymoron – could not have envisaged David Moyes's team sitting fifth in the Premier League and Thierno Barry just behind Erling Haaland in the goalscoring chart at the end of the club's calamitous transfer window. Yet there they are, team and striker defying all expectations, following a rousing if fortuitous comeback against luckless Coventry.

Frank Lampard endured a torturous night against his old club as Coventry had two strong penalty shouts dismissed while the second of Barry's double, taking him to 10 goals for the season in all competitions, appeared to come off his arm. Kiernan Dewsbury-Hall sealed Everton's third consecutive win in convincing style late on.

It was a landmark occasion for Moyes as he equalled the great Harry Catterick's record of 592 games in charge of Everton. "Seeing my name alongside Harry Catterick's is hugely humbling and I hope I can continue to add to that figure," the Scot wrote in his programme notes. "What I really hope, though, is that I can bring trophies to the club like Harry Catterick did." Equalling Catterick's collection of two league titles and an FA Cup may be beyond Moyes, who is in the final year of his contract and working in very different times, but what he is doing with the smallest squad in the Premier League ranks as an achievement in itself.

Coventry made a positive start on their first visit to Hill Dickinson Stadium before Everton began to dominate possession and control the tempo, albeit without seriously troubling Carl Rushworth in the visiting goal. Kiernan Dewsbury-Hall, fresh from being overlooked again for an England squad, and Tyrique George combined well to create a decent early opening for Thierno Barry. The striker took too long over his shot and was eventually crowded out, although Vitaliy Mykolenko was offered a second bite when Jay Dasilva's clearance deflected into his path. Rushworth saved the full-back's attempt from 18 yards out.

The early threat proved a false dawn as far as the hosts were concerned. Everton's only other chance in the first half fell to Harrison Armstrong from a Dewsbury-Hall free-kick. The midfielder's first attempt was smothered by Rushworth and his second, a close-range header, was blocked on the line by Victor Torp but Barry was eventually penalised for handball in the buildperformance became increasingly laboured. Coventry, coping comfortably, finished the first half on top.

Tatsuhiro Sakamoto forced Jordan Pickford into his first save of the night after being released inside the Everton area by Jack Rudoni's slide-rule pass. James Tarkowski then survived two penalty appeals when leaping between Caleb Yirenkyi and Ethan Pinnock in an attempt to clear a Coventry free-kick. The Everton captain appeared to pull the former's hair and raise an arm towards the latter's face. Neither referee Stuart Attwell nor the video assistant referee, Timothy Wood, were minded to award a spot-kick.

Rudoni's influence grew as the half progressed and, after Mykolenko had been dispossessed deep inside his own half, the Coventry midfielder shot just wide of Pickford's right-hand post from distance. Torp followed suit with a free-kick from 30 yards that left the Everton goalkeeper scrambling across his line to save.

There was a smattering of boos from home fans when Everton trudged off at half-time. Their disgruntlement should also have been laced with relief over the Tarkowski penalty decision. Last season the Everton defender could have been sent off – as Michael Keane was for a similar offence against Wolves – and could have conceded a penalty. This season there has to be excessive force for a hair-pull to result in a foul and Tarkowski's action did not meet the threshold.

Coventry had a second strong penalty appeal dismissed moments after the restart when Taiwo Awoniyi went down under a challenge from Mykolenko. The Everton defender was adjudged to have made contact just outside the area.

The visitors finally, deservedly, got a break when benefiting from some abysmal Everton defending to take the lead. Rudoni was again involved with a pass into Yann Gboho who, with the aid of a deflection off Tarkowski, easily held off Merlin Röhl, Harrison Armstrong and Jarrad Branthwaite before poking a shot past the unsighted Pickford.

Less than 10 minutes later, and to general astonishment, Lampard's team were behind. Tyler Dibling had been quiet on his first league start of the season but was involved in the equaliser with a dangerous cross from the right. Coventry failed to clear and when Mykolenko swept home a cross at the second attempt Barry, ghosting between Pinnock and Dasilva, beat Rushworth with an emphatic header.

The French forward soon reached double figures for the campaign when bundling home George's delivery to the far post. Dasilva made a mess of his attempted clearance and headed the ball against Barry – his hand, possibly – before it spun past Rushworth and over the line.

There was no questioning the legitimacy of Barry's next goal involvement. The confident striker split the Coventry defence with a superb pass into Dewsbury-Hall, who ended his disappointing day on a high by racing clear and beating Rushworth with a fine finish inside the goalkeeper's near post.