The Foxes dominated this one-sided clash from start to finish and could even have won it by a wider margin.
The visitors made a dream start with a goal after only three minutes. Lively forward Matty Fryatt burrowed his way into the box, and when his initial shot was saved by goalkeeper Shane Higgs, he reacted quickest to retrieve the ball by the byline.
Fryatt pulled it back for midfielder Lloyd Dyer to lash a shot into the roof of the net from a tight angle.
Cheltenham's Craig Armstrong had a shot kicked off the line by defender Kerrea Gilbert two minutes later but Leicester had chances to increase their lead over the next 20 minutes.
Higgs dived to parry shots from Fryatt and Matt Oakley; and first Fryatt and then midfielder Andy King went close with efforts from the edge of the area.
Cheltenham suffered a further setback when target-man Paul Connor was forced off with a knee injury but they created a couple of chances in the lead up to half-time, visiting goalkeeper Dave Martin having to react sharply to deny Jennison Myrie-Williams and Ashley Vincent.
After the break it was a one-sided affair with Leicester increasing their lead on 54 minutes. Fryatt got away down the left, cut inside and pulled the ball back for captain Matt Oakley, whose low right-foot shot from 22 yards skimmed inside Higgs' right-hand post.
Fryatt scored a deserved goal on 77 minutes when he blasted a 20-yard free-kick seemingly through the defensive wall and Steve Howard, who had been fouled on the edge of the area to set up that chance, saw a goal disallowed for a foul on Michael Townsend shortly afterwards.
The final goal arrived ten minutes from the end when the pacy Dyer raced onto a long ball, comfortably out-paced full-back Jerry Gill and fired a shot past Higgs from the edge of the area.