The Foxes have now collected 11 points from six league games since the Frenchman's appointment as Craig Shakespeare's successor, losing only to leaders Manchester City.
Leicester were in front early in beating Tottenham 2-1 on Tuesday and the hosts made a similarly strong start here by scoring the opening goal after six minutes at the King Power Stadium.
A Riyad Mahrez cross evaded the entire Burnley back line, with goalkeeper Nick Pope pushing the ball straight out to Gray, who tapped in his second Premier League goal of the season, though his momentum saw him clatter into the post.
Burnley, who had won four of their last five Premier League games, saw their attacking potency severely restricted by a serious-looking injury suffered by Robbie Brady, who was taken off on a stretcher, although Johann Gudmundsson wasted a fine second-half chance to level.
Leicester's revival sees the 2015-16 champions remain ninth but narrow the gap to those above them, having extended a 10-year unbeaten home record in the league against the Clarets, who drop a place to seventh.
Following a scrappy start that saw Brady have a weak penalty claim rejected, Leicester took the lead.
Mahrez was given too much space on the right wing, the Algerian cutting inside on to his left foot to deliver a dangerous cross that Pope could only divert straight to Gray for a simple finish.
The Leicester winger collided with the post as he turned the rebound in, but was not hurt in the process of scoring the goal and remained on the field.
James Tarkowski headed Brady's right-wing corner just wide and Gudmundsson cushioned a volley straight at Kasper Schmeichel as the visitors rallied.
Brady left the field on a stretcher with his left leg in a brace, having clattered into Harry Maguire in the process of shooting on the edge of the box, then Pope had to beat away a Mahrez drive away after 42 minutes as Leicester threatened on the break.
Ben Chilwell whipped a first-time drive narrowly wide from an inventive Leicester corner, before a brilliant Phil Bardsley block prevented Wilfred Ndidi from tucking home.
Bardsley cleared off the line as Leicester almost bundled in a second but Gudmundsson should have levelled at the hour.
The Iceland international drilled an effort into Schmeichel's legs after racing on to a Jeff Hendrick pass and holding off the challenge from Wes Morgan, with more composure required from the winger.
Pope tipped a sharp Marc Albrighton strike around the near post after 69 minutes but the goalkeeper was left helpless as Gray beat him with a placed effort that clipped the outside of the upright.
Jamie Vardy was seeking his 50th Premier League goal but Pope denied the England striker as he tried to flick in a teasing cross from the impressive left-back Chilwell.
Sean Dyche threw on strikers Ashley Barnes and Sam Vokes and both substitutes missed chances from presentable crosses as the visitors slipped to their second away loss of the league season as Puel's Leicester march on.
Key Opta facts:
- Leicester are unbeaten in their last seven league home games against Burnley (W4 D3 L0), with wins in each of the last two.
- The Foxes have lost just one of their last 10 games in all competitions (W5 D4 L1), losing 0-2 against Manchester City.
- This is just the second time in their last 11 Premier League away games that Burnley have failed to score, also drawing a blank at City in October.
- Demarai Gray has found the net twice in 14 Premier League appearances this season, double the total he had in 30 appearances last term (1).