The former Southampton boss, appointed on Wednesday to replace the sacked Craig Shakespeare, saw Jamie Vardy finish a sweeping counter-attack and Demarai Gray benefit from a Jonjoe Kenny slice into his own net.
“I saw a team in Leicester with a good attitude, a good structure and a lot of solidarity,” said the Frenchman, whose team climbed to 11th in the Premier League.
“I was impressed with my players. I hope it’s just the beginning of something.”
Everton, under caretaker David Unsworth after Ronald Koeman’s sacking on Monday remain in the bottom three.
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 Shakespeare his job.
The 2015-16 Premier League champions have now won three in a row in the top flight and Carabao Cup – two under caretaker boss Michael Appleton and this maiden success for Puel
