The Foxes had to do without the influential Algerian after he failed to report for training on Tuesday and did not travel to Merseyside with the rest of the first-team squad.
And Leicester suffered their first defeat in seven games and their first of the calendar year despite Jamie Vardy's second-half penalty halving Everton's lead after Theo Walcott's first-half brace.
Kelechi Iheanacho struck the woodwork twice in the space of a few seconds while Matty James had a shot cleared off the line but Leicester could not force an equaliser at Goodison Park.
And Puel felt that their response to Walcott's opener on 25 minutes, having started brightly on Merseyside, was their ultimate downfall.
Leicester "deserved a draw" says Puel
"It was a tough game," admitted Puel afterwards. "I think we started well in this game with quality and we won the duels and the battles against this strong team who played long balls, second balls and strong pressing.
"We gave a good response until the first goal conceded. After this first goal we lost our calm and the control of the game. The opponent took confidence and put us under pressure.
"We came back in the second half with desire, good character and good personality to come back into the game.
"We hit the bar three times in the game. At the end, after the second half, we deserved a draw I think.
"But this game is a good lesson also because in our situation we have to keep the control and the calm and keep the good emotion.
"After the first goal conceded, it was not enough. We have some young players on the team and it's a good lesson for the future."
Foxes had "ingredients and attributes" to beat Everton even without Mahrez
Mahrez was denied his desired deadline day move to Manchester City hours before kick-off with the league leaders baulking at the size of the fee that Leicester demanded for their star man.
But Puel felt that even without the winger, Leicester had enough quality to produce a better performance than they did against Sam Allardyce's men - who had not won any of their previous seven.
Asked about the effect of Mahrez's attempts to force a mid-season exit, the Frenchman continued: "We didn't look for some excuse about the result tonight.
"We had all the ingredients and all the attributes to play a good game and to give a good response together. It was not enough tonight.
"We have to put it right for the next game and to take all the different situations of this game with good control and analyse this with calm and with good desire to react together."