Lloyd Dyer opened the scoring for the Foxes early on after Robins goalkeeper Shane Higgs parried Matty Fryatt's drive into the path of Dyer, who fired high into the net from close range.
The visitors continued to dominate possession and doubled their advantage after half-time when Matt Oakley's 25-yard effort flew past Higgs and into the bottom corner.
Oakley went close again but his volley was well parried by Higgs, who was then tested by Andy King's long-range effort but acrobatically tipped the ball over the bar.
Fryatt added a third with 13 minutes remaining, curling in a free-kick in from 25 yards after Steve Howard was fouled outside the area.
Dyer added his second and completed the rout for Nigel Pearson's side 10 minutes from time when he cooly slotted home from 12 yards.