Beckford struck at the beginning of each half to take his tally to 14 goals in as many games in the competition.
Swindon took their time to settle but once they did they played the type of football that has made them League Two promotion contenders under Paolo Di Canio and Foxes goalkeeper Kasper Schmeichel was called upon to make a handful of decent saves in the first half.
In the end though, Leicester's superior know-how came to the fore to ensure there was no upset at the King Power Stadium.
Paul Gallacher overcame the back problem that forced him off just before half-time against Southampton as Foxes manager Nigel Pearson named the same side that started the 2-0 victory at St Mary's in the Championship on Monday night.
Swindon made just one change with defender Alan McCormack restored to the starting XI on his return from suspension in place of Aden Flint.
Thirty-nine places separated the two clubs in the Football League prior to kick-off and for the first 20 minutes it told.
Beckford put Leicester ahead after five minutes as Swindon made a nervy start.
Foxes captain Matt Mills got the ball into the penalty area following Gallacher's corner and Beckford made the Robins pay for some poor marking by striking a half-volley past Wes Foderingham from 12 yards out.
When Swindon finally did settle it took a superb save by Schmeichel to keep out Oliver Risser's header from Matt Ritchie's cross.
Schmeichel was called upon again just before the half-hour mark to parry Alessandro Cibocchi's shot from the edge of the box.
Ritchie thought he had equalised soon afterwards but a last-ditch challenge by Sean St Ledger deflected his six-yard effort over the crossbar.
Di Canio's side continued to put Leicester under pressure and worked Schmeichel again on the stroke of half-time when the former Manchester City trainee had to push away Cibocchi's long-range free-kick.
However, the League Two club, who were much less of a threat in the second half, were left rueing their missed opportunities eight minutes after the break when Beckford bagged his and Leicester's second goal.
Paul Caddis tried to cut out Gallacher's cross from the right but the ball broke to Beckford who scored with a rising shot from just in front of the penalty spot.
St Ledger came close to making it 3-0 after an hour when he headed wide from Gallacher's corner.
Ritchie responded for Swindon midway through the second half but his 18-yard shot cleared the crossbar.
Another Gallacher cross soon afterwards almost brought a goal for David Nugent but his close-range header was brilliantly saved by Foderingham.