Everton 5 Huddersfield Town 1: Beckford off the mark for Everton as Town crash out
The former Leeds United striker – jeered all night by the travelling supporters – scored a 50th-minute penalty just as Town sensed the possibility of a comeback.
It was cruel luck on Lee Clark's men who had grabbed a lifeline just before the break and started to ask questions of the Premier League club.
As if rubbing salt in the wounds, Beckford then laid on the pass for substitute Louis Saha to score Everton's fourth.
The margin of victory could have been even greater if Beckford had not allowed team-mate Johnny Heitinga to take Everton's second penalty of the night, which was duly saved by substitute goalkeeper Ian Bennett after 71 minutes.
Defender Lee Peltier was sent off for the challenge on Seamus Coleman – his second bookable offence, but Leon Osman's late header still made it 5-1 to the Toffeemen, underlining the gulf in class.
Town, who had completed the signing of Crystal Palace striker Alan Lee yesterday morning, were two goals down after just 14 minutes . Yet winger Gary Roberts – a lifelong Evertonian – dragged his side back into the contest just before half-time.
Everton, who finished eighth last season in the Premier League, were the highest-ranked side in this stage of the competition but had rested six players from the weekend.
Only defenders Sylvain Distin, Heitinga, and Leighton Baines, winger Steve Pienaar, and striker Jermaine Beckford survived the squad rotation.
Town manager Clark made four changes – presumably with one eye on the important League One game against Charlton Athletic this weekend.
Both Jordan Rhodes and Joe Garner dropped to the bench while midfielder Kevin Pilkington was absent after taking a knock at Peterborough.
Naysmith returned from suspension and replaced Graham Carey while Theo Robinson, a surprise inclusion, played as a lone striker in a 4-5-1 formation.
Although Jamie McCombe tested Everton's debutant goalkeeper Jan Mucha with an early header, it took just seven minutes for the home side to score through Marouane Fellaini.
The towering Belgian arrived unmarked at the far post, met a cross from Magaye Gueye, and sent a header into the roof of the Town net.
When Jack Rodwell doubled Everton's lead seven minutes later – striking a low drive beyond the reach of Smithies – the scent of a landslide home victory was hanging in the air.
Clark started to show signs of frustration in the away technical area while the 5,100 travelling supporters briefly lost their voices, although they could still raise a jeer for Beckford.
Without a goal in his first two appearances since a summer move from Town's local rivals Leeds, Beckford shrugged off the stick and was always on the look out for the ball.
At 2-0, Everton may just have been feeling the game was over. If that was the case, Town had other ideas.
Roberts had achieved a boyhood dream simply by playing on the Goodison Park pitch for the first time, but his evening got a whole lot better when he scored five minutes before the break.
Released by Liverpool as a schoolboy, Roberts initially turned a cross from Joey Gudjonsson onto the Everton bar from close range. The ball appeared to be dropping over the line when Heitinga, under pressure from Robinson, put a head on it. Roberts wheeled away in celebration, leaving no-one in any doubts about who was claiming it.
Town changed goalkeepers at the interval with veteran Bennett coming on for his debut as a replacement for Alex Smithies.
But Bennett was soon picking the ball out of the net as Beckford restored Everton's two-goal lead with a penalty following a foul by Naysmith on Rodwell.
Buoyed by the goal, Beckford nearly grabbed a second but Bennett tipped over his angled drive. New signing Lee had completed his move to Town yesterday morning but was cup-tied last night.
Lee, 32, signed a two-year deal and should make his debut in Saturday's home game with Charlton.
The fee for the former Rotherham United and Republic of Ireland international has not been disclosed but was actually agreed in the summer and is likely to contain promotion- related incentives.
Chief executive Nigel Clibbins said: "We will not disclose the detailed terms but suffice to say that the speculation regarding a three-year deal, 500,000 transfer fee, and wages which emanated from quotes in the London media are once again hugely exaggerated. In fact, the initial outlay and the structure of the deal represent excellent business for the club."
Everton: Mucha; Coleman, Heitinga, Distin, Baines (Hibbert 81); Fellaini; Pienaar (Saha 75), Osman, Rodwell, Gueye (Bilyaletdinov 65); Beckford. Unused substitutes: Jagielka, Arteta, Cahill, Howard.
Huddersfield Town: Smithies (Bennett 45); Peltier, McCombe, Clarke, Naysmith (Kay 67); Croft, Johnson, Gudjonsson; , Arfield; Roberts; Robinson (Novak 75). Unused substitutes: Garner, Rhodes, Goodwin, Carey.
Referee: ND Swarbrick (Lancashire).