It did not take long for the home side to exert their influence on the match and they led after barely five minutes.
Skipper Matt Mills found the in-form Beckford and the former Everton man made no mistake from inside the area.
Leicester continued to dominate possession in the opening stages but Swindon were so nearly level after 30 minutes when Alessandro Cibocchi found space in the area, but Kasper Schmeichel was on hand to push the ball to safety.
Cibocchi was again involved just before the break and might have levelled for Swindon with a well-executed free kick, but Schmeichel was again on hand to remove the danger.
The home side added their second of the match barely five minutes into the second period when Paul Gallagher found Beckford, and the striker added his second of the match with a finish from close range.
Matt Richie went close for the away side after 72 minutes with an effort inside the area, but it drifted wide when he might have done better.
The home side controlled the latter stages to continue their recent run of solid form.
Leicester resume domestic duties against Middlesbrough on Wednesday, while Swindon travel to Southend a day earlier.