Diomansy Kamara starred on his home debut, opening the scoring in the 23rd minute and winning the penalty from which Gallagher made it 2-0.
The Scot made it three with a stunning long-range strike and Patrick Van Aanholt made it four to secure the Foxes a second win in eight matches and move them into the top 10 in the Championship table.
Leicester boss Sven-Goran Eriksson was forced to reshuffle his defence, with Jeffrey Bruma suspended and Kyle Naughton missing out through illness.
Miguel Vitor and Ben Mee came in at the back, with Andy King returning to the centre of midfield.
Burnley made four changes from the side beaten 2-1 at home by Ipswich last time out, with Brian Jensen, Wade Elliott, Chris Iwelumo and Michael Duff all starting.
Kamara was inches away from connecting with a Gallagher cross after just two minutes.
At the other end, Nathan Delfouneso tried his luck with a shot from 25 yards which took a deflection that had Foxes goalkeeper Chris Weale scrambling.
The lively Kamara sent a header over the crossbar from a Van Aanholt cross shortly before the on-loan forward did open the scoring.
Richie Wellens played a neat pass through for Ayegbeni Yakubu who turned Duff superbly and was shaping to shoot, only for Kamara to nick the ball off his foot and drill home.
Kamara was a constant menace, and Jay Rodriguez was booked for bringing him down with a reckless challenge.
The Clarets started the second half brightly, with Mears somehow clearing the crossbar after Weale had spilled a Delfouneso cross.
But it was the home side who extended their advantage in the 51st minute when Kamara was brought down by Andre Bikey in the box and Paul Gallagher smashed home the penalty off the body of Jensen.
Wellens sent Yakubu clear from the Foxes' next attack but he failed to get his shot off as Duff put in a superb challenge.
It was all Leicester, and they added a third after 71 minutes. Gallagher picked up the ball in acres of space and raced forward before unleashing an unstoppable 25-yard drive into the bottom corner.
Van Aanholt scored his first-ever league goal for City to make it 4-0 with nine minutes remaining. The Dutchman exchanged passes with King before curling the ball past Jensen.
There was still time for Wellens to slam a shot against the crossbar and Jensen to deny Gallagher a hat-trick with a fine diving stop as the rampant Foxes finished strongly.