They were largely restricted to long-range efforts, but probably should have had a penalty when Christian Fuchs brought down Aaron Lennon soon after Leicester’s second goal.
Though Everton dominated the second half, a lack of incision was indicative of a team that has scored only seven league goals this season and are winless in seven games in all competitions.
Leicester, meanwhile, have staged a turnaround since a run of six league games without a win cost former manager Craig Shakespeare his job.
The 2015-16 Premier League champions have now won three in a row in the top flight and EFL Cup – two under caretaker boss Michael Appleton and this maiden success for Puel.
New top dog, old tricks
Though Puel guided Southampton to the EFL Cup final last season, his appointment was greeted by derision in some quarters, especially after the Saints managed just 41 league goals last season.
In claiming victory on his bow at the King Power Stadium, Puel saw his Leicester side employ the method that brought them a shock title success little more than a year ago.
Everton had more of the ball, spent more time in the Leicester half and had more players in advanced positions – and yet the home side had the greater threat going forward, especially through their trademark counter-attacks.
Gray, in his his first league start for a month, was excellent, and it was his electrifying pace that created Leicester’s first goal.
An Everton free-kick was headed clear, with Gray collecting the ball deep in his own half. He evaded three challenges and fed Riyad Mahrez, whose cross was finished at the back post by Vardy.
The second goal was tinged with fortune as Gray’s cross-shot from the left was sliced past Everton keeper Jordan Pickford by the unlucky Kenny.
The home side were under pressure for most of the rest of the contest, but a back four led by Wes Morgan rarely looked like being breached.
A big job for someone
Unsworth has stated his desire to be given the reins at Goodison Park full-time and he stamped his mark on the Toffees by relegating £45m summer signing Gylfi Sigurdsson to the bench and giving first league starts of the season to wingers Lennon and Kevin Mirallas.
The early problems, though, were at the back, as the central defensive partnership of Phil Jagielka and Ashley Williams, with a combined age of 68, were exposed by Leicester’s pace.
By the time they showed anything positive, Everton were two down.
Lennon twice found space down the right – he probably should have take a shot at goal instead of passing to Dominic Calvert-Lewin, who missed his kick, and was unfortunate not to win a penalty when Fuchs’ challenge failed to make any contact with the ball.
Mirallas also had two efforts on goal, so it was surprising when both wide men were hauled off at half-time, with Oumar Niasse joining Calvert-Lewin in attack.
Everton pressed, spending most of the second half in Leicester territory, but their only chances of note were a Jagielka overhead kick and a Leighton Baines effort that dipped over the bar from long range.
Whoever is given the Everton manager’s job will inherit a side that has only scored more goals than Bournemouth and Crystal Palace, conceded more than all but Palace and not won any of their past 13 away league games.
Man of the match – Demarai Gray
Vardy starts the party – the stats
The Foxes have taken six points in their last two Premier League games, the same number they managed in their opening eight matches of the season.
Jamie Vardy has been involved in 13 goals in his last 15 Premier League games at the King Power Stadium (11 goals, 2 assists).
Vardy has scored the first Premier League goal under each of the last three Leicester managers (Claudio Ranieri, Craig Shakespeare and Claude Puel).
Riyad Mahrez has assisted nine of Jamie Vardy’s Premier League goals – more than any other player.
Everton named eight English players in their starting XI, the second time they’ve done so in the space of nine Premier League games (also against Man City); prior to this, they hadn’t done so in any of their previous 780 Premier League games, since naming eight in the starting XI against Manchester United on Boxing Day 1997.
The last four managers to take charge of Leicester in the Premier League have all won their first matches in charge (Ranieri, Shakespeare, Appleton and Puel).
Everton have now conceded 20 goals in the Premier League this season, their most after 10 matches of a league season since 1994-95 (22).
The Toffees have used six teenagers in the Premier League since the start of last season, two more than any other club (Ademola Lookman, Mason Holgate, Nikola Vlasic, Dominic Calvert-Lewin, Tom Davies and Beni Baningime).
‘I hope it’s just the beginning’
Leicester manager Claude Puel: “I was impressed with my players. It was a very good first half and there were good combinations between the players for the first goal.
“It was more difficult in the second half because Everton are good, but I saw a team with a good attitude, good structure and solidarity.
“It was a day that was perfect but I hope it’s just the beginning of something.”
Everton caretaker boss David Unsworth: “The second half was much better. We gave them a two-goal start – in the first half we didn’t perform. Our display was much better after the break.
“The best tactics I thought was with a diamond and two strikers. Aaron Lennon did OK in the first half and Kevin Mirallas couldn’t get into the game. I wanted to be brave and positive in the first half.
“I went with the best team to try to win a game of football. The first half wasn’t acceptable, but the second half we didn’t get the rewards that our dominance justified.
“Whoever gets the honour of being Everton boss, including me, needs time with this set of players.”
Everton travel to Lyon on Thursday (18:00 GMT) knowing that defeat will probably put them out of the Europa League.
The Toffees are back in Premier League action at home to Watford next Sunday (16:30), with Leicester at Stoke the day before (12:30).