City have now also taken 14 points from their last seven games since sacking Craig Shakespeare, with their latest three-point haul - as slender as it was - giving new boss Claude Puel plenty to smile about.
Leicester, who had gone 10 matches without defeat to Burnley prior to their most recent encounter earlier this year, soon started growing back into the game and nearly had a second through a well-struck Mahrez drive.
The Foxes could not kill their opponents off, though, as Phil Bardsley produced a fine block to prevent Gray from doubling his own personal tally, before blocking Wes Morgan's prodded effort off the line.
Schmeichel was also having to remain alert at the other end of the field, keeping out another Gudmundsson attempt as the visitors attempted to maintain their impressive recent form.
The post then denied Gray at the other end, shortly after Marc Albrighton tested Pope, but the final big chance of the afternoon would fall the visiting side's way.
Gudmundsson's cross was crying out for Ashley Barnes to convert, though he missed his kick and Leicester swept up the danger for three more points.
Leicester City: Schmeichel; Simpson, Chilwell, Morgan, Maguire; Iborra, Ndidi, Gray (Okazaki 88'), Albrighton; Mahrez, Vardy
Burnley: Pope; Mee, Tarkowski, Ward, Bardsley; Gudmundsson, Cork, Brady (Arfield 31'), Defour (Barnes 66'); Hendrick; Wood (Vokes 74')