The self-proclaimed Special One has heaped praise on Hazard for his improved work ethic this season but was critical of the 23-year-old's lax marking in the Champions League semi-final second-leg loss to Atletico Madrid.
"Next season we want more from you," said Mourinho, before adding "you win the kid (Hazard)" and handing over the trophy at the Stamford Bridge awards ceremony on Monday night.
With tongue in cheek, Hazard, chosen ahead of Gary Cahill, Cesar Azpilicueta and John Terry for the award, had words of consolation for his overlooked team-mates.
Hazard, who scored 17 goals this term, said: "Maybe next season, you have to work a bit more."
Mourinho's rebuke of Hazard led to suggestions the forward may be allowed to depart this summer, with Paris St Germain among the interested parties - but Hazard insists he is at home in London.
"It was a great season for me and I'm happy to play for Chelsea," he added. "I'm enjoying this city. I live in Cobham (near Chelsea's Surrey training base). It's easy."
Hazard was Chelsea's sparkling performer this season, with much of the onus on him in attack as the Blues' strikers lacked conviction in front of goal.
Mourinho has vowed to spend the summer planning and improving his squad, which he insists does not require major surgery, although a prolific goalscorer is the number one target.
The Blues boss has vowed to take time to consider changes to his squad and denied that his preference is for transfer dealings to be completed prior to the World Cup.
Rumours persist of Chelsea landing Brazil-born Spain striker Diego Costa in a £32million deal - matching his release clause - from Atletico Madrid.
Atletico's involvement in the Champions League final on May 24 means the deal may not be formerly concluded and announced until after the Lisbon showpiece for a player who was shackled by Chelsea's defence in the two-legged semi-final, scoring only from the penalty spot.
And, perhaps in some way to finance the deal, Brazil defender David Luiz is reportedly close to a switch to Barcelona after the Catalans appealed their transfer ban to allow them to sign players.
Mourinho is determined Chelsea begin next season, his second on his return, with the Premier League title as the target.
"It's good to be back but it's not good to finish third," Mourinho added.
"I'm happy with the evolution, but I'm not happy with third. Next season we want to do better than we did.
"We have to do better, I have to do better, the players have to do better.
"I have great belief in them and I think next season we can start day one and we can say without any kind of fears that we are there not to fight for the title, to win it."