Beckford's first competitive goal for the Toffees, coming from the penalty spot, helped put the seal on the victory after Huddersfield had responded to early Marouane Fellaini and Jack Rodwell strikes.
A John Heitinga own goal gave the League One outfit hope at half-time but Beckford's penalty and further efforts from Louis Saha and Leon Osman - after Lee Peltier had been sent off and Heitinga missed from the spot - settled the tie.
Everton boss David Moyes fielded a strong side despite making six changes.
Midfielders Jack Rodwell and Fellaini both came into the side at Goodison Park after being named on the bench for Saturday's draw with Wolves.
Goalkeeper Jan Mucha and forward Magaye Gueye were handed debuts while Osman and Seamus Coleman also featured.
Huddersfield's side included former Everton defender Gary Naysmith, who spent seven years with the Toffees before leaving in 2007.
Everton had a chance after winning a corner inside the first minute but Fellaini miscued his shot from Gueye's cross.
Jamie McCombe had a glorious chance to hand the Terriers the lead in the seventh minute when Lee Croft's cross fell invitingly after taking a looping deflection off Leighton Baines.
The summer signing from Bristol City met the ball with a firm header but Mucha produced a fine save to his left.
Everton responded to claim the opening goal with a slick move of their own.
France Under-21 international Gueye found good space on the left and delivered a pinpoint cross for the unmarked Fellaini to head in from six yards.
The Toffees began to look a class apart and stepped up the pressure to add a second seven minutes later.
Rodwell beat Damien Johnson with a neat drag-back on the edge of the area and then drilled a low shot into the bottom corner.
Yet Huddersfield were determined to put up a fight and Peltier blazed over from 18 yards.
Gary Roberts also had a chance from a well-worked Johnson free-kick but shot well over the bar.
Huddersfield then survived a scare as Sylvain Distin got forward and whipped in a dangerous cross which McCombe nervously sliced across his own goal.
Steven Pienaar then drove narrowly wide after Fellaini smartly pulled the ball down in the area.
But Huddersfield finished the first half strongly and twice hit the woodwork before pulling a goal back.
First Croft struck the foot of the post from a Naysmith cross.
Town then came back again with a Joey Gudjonsson cross bouncing off Roberts and hitting the bar.
Robinson closed in the rebound and Heitinga headed into his own net under pressure.
Huddersfield suffered a blow as they were forced to replace their highly-rated goalkeeper Alex Smithies, who was named in the England Under-21 squad yesterday, with Ian Bennett at half-time.
One of Bennett's first jobs was to pick the ball out of the net after Beckford celebrated his first goal for Everton.
The former Leeds striker stepped up to score from the penalty spot in the 49th minute after Rodwell was brought down by a clumsy challenge from Johnson.
Beckford almost had a second moments later after Rodwell charged down a clearance but Bennett produced an instinctive save to tip over the bar.
Rodwell then shot tamely at Bennett after tricky work from Fellaini on the edge of the box and Pienaar missed the target from distance. Osman then went even closer with a fine curling effort.
Croft was denied at the other end by the feet of Mucha but Everton continued to look dangerous and Fellaini and substitute Diniyar Bilyaletdinov had shots blocked in a goalmouth scramble.
Everton did then bundle the ball into the net but the flag was raised for a foul on Bennett.
Everton were awarded a second penalty after Peltier pulled back Coleman with 20 minutes remaining.
Having already been booked for a foul on Gueye, Peltier was shown a second yellow card by referee Neil Swarbrick and then a red.
Heitinga, still awaiting his first Everton goal, was handed the responsibility from 12 yards but Bennett saved well to his left.
Yet Everton did not have to wait long for their fourth, Saha coming off the bench to score with his first touch from 18 yards after Osman and Beckford combined.
Osman then wrapped up the scoring five minutes from time with a header at the far post.