Martin Jol admitted after this absorbing Capital One Cup tie that he was a relieved man – and it was no surprise. Fulham, so fitful in the first half, were rescued by the predatory instincts of Dimitar Berbatov and Darren Bent, who fired Jol’s side to their first home win in five months after Everton had threatened to increase the pressure on the Dutch coach at Craven Cottage. Roberto Martinez’s side seemed well set to extend Fulham’s miserable set to the season after Steven Naismith gave the visitors a deserved twelfth minute lead, but the home side were far more purposeful after a half-time dressing down.

Dimitar Berbatov grabbed his first goal of the campaign after Seamus Coleman’s tackle on Adel Taraabt inadvertently sent the ball into the Bulgarian’s path and, then, just three minutes after stepping off the substitutes’ bench, Bent drove a powerful shot through Joel Robles and into the net from a cleverly-worked free-kick by Giorgos Karagounis. The contrast between Fulham’s feisty second half showing and the limp nature of a forgettable first half was marked. With three potentially winnable league matches against Cardiff City, Stoke City and Crystal Palace to come in quick succession, the question must be whether Jol can accommodate both of his goalscorers in the same side.

Seeking a swift response to the derby defeat at Chelsea on Saturday night, Jol made six changes from the side swept aside at Stamford Bridge. The impressive Elsan Zverotic made his debut at right back, with Philippe Senderos replacing Fernando Amorebieta and John Arne Riise coming in for Kieran Richardson. Karagounis anchored the midfield as Steve Sidwell dropped out, while Adel Taraabt was preferred to Damien Duff down the left and the fit-again Berbatov replaced Bent up front. Roberto Martinez rotated his team, too, but it was Everton who made by far the more accomplished start.

Barcelona loanee Gerard Deulofeu gave Riise a torrid time down the Everton front and the excellent Romelu Lukaku cut a towering figure as he led the Everton attack, frequently posing problems for the Fulham back line both in the air and, more surprisingly, by surging from deep with the ball at his feet. The visitors dominated the first half, with Lukaku shrugging off the attentions of Brede Hangeland early on to set up a chance for Naismith, whose snapshot was smothered by Stockdale. The Scottish international was getting real joy by running across the hosts’ defence from the right wing – and Fulham failed to heed the early warning signs.

A wonderfully floated through ball from Delofeu caused chaos in the Fulham back-line, with the defence pushing up as Taraabt and Scott Parker retreated. Naismith, played onside by Parker, strolled through an alarming gap to steer a calm finish between Stockdale’s legs and give Everton the lead. How Martinez’s side were only a single goal to the good at the break was anyone’s guess. Sylvain Distin struck a post with a powerful header from a corner, before a splendid run from Deulofeu took him past three Fulham defenders only for Hangeland to save the day with a last-ditch tackle. Lukaku lost his composure at the end of a mazzy dribble into the box and snatched horribly at his shot, before Delofeu’s clever lay-off gave Naismith a chance to grab a second, but he shot  into the side netting from a narrow angle.

Fulham’s only real threat came from Berbatov, who sent two headers wide from dangerous positions, and a smattering of boos greeted the half-time whistle. It was a different story in the second half as Fulham attacked from the off, with the pedestrian pace of the first period a distant memory. Swiss midfielder Pajtim Kasami came more and more into the game, drifting into dangerous positions behind Berbatov, and whipping over a succession of threatening crosses from wide positions. A promising run from Kasami led to a powerful drive from Alex Kacaniklic stinging the fingertips of Robles and Fulham’s pressure finally told nine minutes after the break.

A strong challenge from Senderos set Fulham moving forward and Parker’s slide-rule pass looked to have given Taraabt half a yard inside the Everton box, but Seamus Coleman’s sliding tackle saw the ball break kindly instead for Berbatov in oceans of space. The former Manchester United forward needed no second invitation to tuck a first-time finish past Robles. The relief around Craven Cottage was tangible and Fulham poured forward. Jol sent Bent on in search of the vital second and the substitute’s impact was almost immediate.

Just three minutes after replacing Taraabt, Bent gave the watching Roy Hodgson a pointed reminder of his finishing ability by punishing Everton for falling asleep at a set-play. There appeared little danger as Karagounis hovered over a free-kick thirty yards from goal, but the Greek veteran quickly slipped a reverse ball into the path of the unmarked Bent, whose venomous shot beat Robles at his near post from a tight angle.

Fulham’s lead was far from secure – and memories of their late capitulation against West Brom eleven days ago – were stirred by Everton’s prolonged period of late pressure. Stockdale was called upon to make two superb saves with his legs from Lukaku and Coleman in the last ten minutes as Jol’s side reached the fourth round for the first time in five years.

FULHAM (4-2-3-1): Stockdale; Zverotic, Riise (Amorebieta 62), Senderos, Hangeland; Karagounis, Parker; Kacaniklic, Taraabt (Bent 64), Kasami; Berbatov. Subs (not used): Etheridge, Riether, Sidwell, Mesca, Rodallega.

BOOKED: Parker.

GOALS: Berbatov (54), Bent (68).

EVERTON (4-2-3-1): Robles; Coleman, Oviedo, Stones, Distin; Heitinga (Jagielka 68), Gibson (Barry 68); Deulofeu, Naismith (Mirallas 80), McCarthy; Lukaku. Subs (not used): Howard, Baines, Barkley, Jelavic.

BOOKED: Gibson, McCarthy, Stones, Coleman.

GOAL: Naismith (12).

REFEREE: Martin Atkinson (Bradford).