As for Everton, the Merseyside outfit lacked belief in the final third and David Unsworth's side remain in the bottom three of the table after a fifth league defeat in seven outings.
Leicester began on the front foot as Puel looked to make an impact after his arrival in the East Midlands and Gray - who had been introduced for just his second league start of the season - almost marked his recall with a goal after three minutes but the winger dragged his shot from 25 yards just wide of the post.
However, Leicester finally opening the scoring in style, with Gray gliding past Tom Davies, Idrissa Gueye and Wayne Rooney before setting up Riyad Mahrez, whose low pass across goal was drilled into the roof of the net by Vardy.
Towards the end of the first half, Everton came back into the game, with Dominic Calvert-Lewin failing to convert Aaron Lennon's cutback, but Leicester doubled their advantage when Gray's inswinging cross was sliced into the far corner by Kenny.
Leicester's good fortune continued when Everton were denied a penalty by the referee despite Lennon seemingly being hauled down by Fuchs when the winger was about to pull the ball back for a teammate.
Kevin Mirallas sent two long-range strikes just wide of Kasper Schmeichel's post just before the break, but it did not prevent the Belgian from being replaced at half time - along with Lennon - with Beni Baningime and Oumar Niasse coming on in their place.
The alterations led to improvements from Everton, but the visitors lacked a cutting edge and Leicester did not need much encouragement to plough forward in search of a third goal.
Everton were able to get back onto the offensive but like earlier in the half, clear-cut chances were few and far between with Gueye seeing a shot from distance deflected over the bar and Rooney heading well off target from 12 yards.
Unsworth introduced Gylfi Sigurdsson in place of Rooney but rather than the alteration instigating a resurgence, Leicester only looked more likely to cap a fine performance with a third goal.
With less than five minutes left, Vicente Iborra should have got his name on the scoresheet after receiving a pass from Vardy, but the former Sevilla man could not get his feet sorted and eventually poked wide.
Leighton Baines fired a late volley over for Everton but that was as good as it got for the Toffees, who look desperately short of form even under the guidance of interim boss Unsworth.