Steven Naismith, on his 27th birthday, scored the only goal seconds before half-time at Goodison Park and despite Chelsea creating a number of opportunities prior and after they could not find a way through.
Experienced forward Samuel Eto'o had a debut to forget as he missed a couple of chances and was denied a certain goal by a last-ditch tackle from the excellent Gareth Barry, also making his first appearance having joined on loan from Manchester City.
"We didn't score goals. We had chances and chances but we didn't score," said Mourinho.
"If you don't score, what you create means nothing. It is a simple story. Artistic football without goals is not good.
"This was the game when we had 21 shots, nine attempts on target and well before they scored the goal we had easy chances to score.
"I don't think it is a question of sharpness. If the ball to Eto'o from (Andre) Schurrle is sent as a slow pass and Eto'o waits for the ball Barry has time to come.
"If it is a fast ball Eto'o scores with an open goal - I don't think this defeat is about sharpness.
"Perhaps there was not the killer instinct: one chance, one goal. Maybe one day we will have three chances and score three goals and win 3-0."
Mourinho credited Everton's goal - Naismith heading in from close range after Nikica Jelavic had kept alive Leon Osman's far post cross - to mistakes made by his players but he refused to blame them.
"My team is always my team: winning, losing, playing well, playing badly - my team is always my responsibility," added the Portuguese.
"We are not speaking about 10 kids but their mistakes are my responsibility, not theirs.
"The opponent didn't create their goal so it is hard to take but tomorrow is a training day and Wednesday (in the Champions League) is another game and Saturday is Premier League again and you have to look forward and work and go for it."
After three draws victory gave Roberto Martinez his first three points as Everton manager after taking over from David Moyes in the summer.
It was a proud day for the Spaniard, although he felt it was no more than his side deserved after playing well this season without due reward.
"It is difficult to describe because it wasn't the perfect performance from our point of view but when you show that character and all the attributes to be a winning side I thought we were a 10 (out of 10) in that respect," he said.
"I thought the crowd understood we are still adapting and we are trying to play in a different manner and it can be difficult when you play a side like Chelsea.
"We gave two or three (early) chances probably from us not keeping the ball well enough but the way we reacted, took responsibility and defended was immense.
"It is one of those victories which will give us a lot to get the fans to understand it is going to take a bit of time to be as good as we can be but we can still win games in the process.
"As a manager you couldn't be prouder. It gives you a great platform to carry on working and get better.
"Whatever you want to do in football you have to win games. There are many aspects you can work on in training but that confidence, belief and desire comes from winning matches and I thought today was important in that respect.
"Our previous three games were more one-sided and we didn't get the results we deserved.
"Football always give you what you deserve and those players deserve to be unbeaten."
- Sports & Recreation
- Steven Naismith
- Gareth Barry