The Claude Puel era is up and running at Leicester City and any dissenting voices about the Frenchman’s appointment will quickly be silenced if this proves to be a sign of things to come. Inspired by the outstanding Demarai Gray, Leicester looked menacing whenever they attacked, and the wonderful breakaway goal that Jamie Vardy converted to set them on their way to victory against a beleaguered Everton side was arguably the best possible riposte to any concerns about Puel’s playing style.

It was a classic piece of counterattacking and featured a splendid 60-yard run from Gray, who delivered a performance that will leave many wondering why the winger did not feature more often under Craig Shakespeare and Claudio Ranieri. Gray was the best player on the pitch by a distance and was also credited with the second goal – rather dubiously – after Jonjoe Kenny had horribly sliced the England under-21 international’s cross into his own net.

It was that sort of day for David Unsworth, who did nothing to improve his chances of getting the Everton job on a permanent basis as they slipped to a ninth defeat in 13 matches, leaving them third from bottom and facing up to a scenario that was unthinkable when £150m was spent in the summer.

Asked whether Everton are in a relegation battle, Unsworth replied: “I think we have to be honest, we are where we are, if that continues then the answer is yes. At the same time it’s nothing that a couple of back-to-back wins wouldn’t put right. But Sunday’s game against Watford is now massive.”

Nothing went right for Unsworth here. His decision to start with wingers was scrapped at half-time, when he withdrew Aaron Lennon and Kevin Mirallas, and the sight of Wayne Rooney raising his arms in bewilderment when his number came up in the second half, as he prepared to take a corner, rather summed things up for a club that appears in serious trouble.

To add insult to injury for Unsworth he finished his post-match press conference answering a question about Joey Barton’s scathing assessment of him during a radio broadcast from the King Power Stadium. Barton described Unsworth as “a glorified PE teacher” among other things. “What Joey Barton says, I couldn’t care less,” Unsworth said.

Everton have far bigger problems than Barton to deal with right now. Toothless up front and devoid of ideas in midfield they rarely looked like scoring despite dominating possession after a disastrous first half hour in which they conceded twice. Vardy, Gray and Riyad Mahrez caused Everton no end of problems with their pace, trickery and movement, in particular in that devastating early period that blew the visitors away.

With confidence so brittle among their players, and a legitimate goal threat painfully lacking in the absence of a proven centre-forward, it was hard to see Everton finding a response. Unsworth, however, was entitled to feel that Everton should have had an opportunity via the penalty spot when Christian Fuchs clattered into Lennon from behind shortly after Gray’s goal. Andre Marriner, the referee, inexplicably failed to point to the spot. “A little bit of Lady Luck isn’t going for us at the moment,” said Unsworth.

Leicester, in contrast, are smiling again and this was a fine way for Puel to begin his reign. They started at a high tempo and the Frenchman’s decision to bring Gray into the team, giving him only his second league start of the season, was totally vindicated. Mahrez also thrived in a more advanced central role, just behind Vardy, and it was a lovely move in which those three attacking players combined to give Leicester the lead.

Gray was the chief protagonist, picking the ball up deep inside his own half after Leighton Baines’s free-kick was cleared. The winger skirted around Tom Davies close to the touchline, stepped inside Idrissa Gueye and left Rooney trailing in his wake as he tore at the heart of the Everton defence. It would have been easy for Gray to go for goal himself at that point but instead he kept his composure and played a perfectly weighted pass that released Mahrez on the right.

Mahrez looked up and delivered a low centre that Vardy was never going to miss from about six yards out. It was a brilliant goal that highlighted the blistering pace in one team and the total lack of it among their opponents.

Leicester’s second owed much to good fortune as Gray, drifting out onto the left, delivered an inswinging cross that looked harmless enough until Kenny wildly miskicked, sending the ball spinning backwards, beyond Jordan Pickford and into the far corner of the net. It was a cruel moment for the 20-year-old right-back, who was making only his second Premier League start, and put Leicester in a commanding position.

Everton switched to a midfield diamond after the interval as Unsworth introduced a second striker before eventually turning to Gylfi Sigurdsson, the club-record signing, as a replacement for Rooney in the 74th minute. That neither Rooney nor Sigurdsson acknowledged one another during that change said it all on another chastening afternoon for Everton.