City's first goal arrived on 37 minutes when Matt Fryatt cleverly beat two defenders on the right before crossing for Lloyd Dyer to score with a crisp drive from 13 yards.
Chris Day kept Boro's hopes alive when, within moments of the restart, he got down sharply to parry a Fryatt shot.
And then the keeper came to his side's rescue again with a spectacular save to keep out Matty Oakley's fierce drive.
But Day was beaten on 52 minutes when Dyer raced down the left before finding Fryatt, who found the net with a carefully placed shot from just inside the area.
Leicester wrapped up the game on 69 minutes when Andy King spotted the keeper off his line and scored with a wonderful chip-shot which went in off the bar from 25 yards.
There could have been more for the home side. The Foxes felt they should have been awarded a penalty on 15 minutes when Eddie Anaclet appeared to have hauled down Fryatt inside the box.
Then on 64 minutes, Fryatt was sent sprawling again inside the area, this time after a clumsy challenge by Mark Albrighton, but again referee Darren Drysdale declined the home side a spot-kick.
Stevenage had their moments after starting brightly.
Aleksandar Tunchev's under-hit back-pass almost let in Steve Morison, but Paul Henderson was alert to the danger and was quickly off is line to clear.
Then on 13 minutes, Anaclet found Morison down the right inside channel but the Stevenage striker missed the target when he dragged his angled shot just wide of the far post.