The Black Cats had dragged themselves back into the game as Jermain Defoe and Steven Fletcher scored to cancel out the hosts' 2-0 lead given to them by Gerard Deulofeu and Arouna Kone.
However, the openness of the game lured Sunderland in and when Allardyce wanted them to close things down to take away a point his players went looking for a winner.
Everton picked them off with Kone scoring twice to complete his hat-trick after a Sebastian Coates own goal and one from Romelu Lukaku had already taken the result away from them
"I am obviously concerned about the lack of understanding in certain periods of the game with the team, particularly when we got back to 2-2," said the Black Cats boss.
"There was nothing wrong with what we had been doing up to then apart from we gave a very soft first goal away which could easily have been avoided
The second goal was a cracker from Kone.
"The (Defoe) goal before half-time and the chances and opportunities greater than Everton's we created in the first half gave me some encouragement to say if we get the second then keep it tight but when we got the second we decided to go and attack for the third.
"We didn't get it and within seven minutes we tossed the game away and allowed Everton to score three silly, sloppy goals from counter-attacks they didn't have to work very hard for.
"That lack of discipline and understanding has really concerned me on the basis we should have been satisfied with the 2-2.
"We should have frustrated Everton and maybe we might have got on the end of a counter-attack and made it 3-2.
"We had 17 attempts on goal and 12 on target
At the top end we can score a goal and create chances but until we get into the habit of defending better out of possession it is worthless how many goals we score if we defend like that.
"That lack of discipline and organisation as a team is obviously the biggest concern for me."
The result was Everton's biggest in eight years since they beat Sunderland 7-1 at Goodison Park.
Kone's first hat-trick in English football was just reward for the players' perseverance after a serious knee injury affected the first two years of his career at the club having followed Martinez from Wigan.
The Ivorian, who was booed by his own fans when he came off the bench in the opening game of the season, was due to be out of contract in the summer but he has now triggered an appearance-based clause which entitles him to extend for a further 12 months.
"It has been as difficult as you can get
It was a career-threatening injury," said Martinez.
"His attitude, his focus, never changed and he never chose to blame anyone and anything.
"Against Watford, in difficult circumstances, it was a real turning point and from that point on he has been very impressive.
"Today was the most complete 90 minutes he has had and it was a phenomenal 90 minutes because it triggers an extension to his contract.
"He is a sensational example to any young footballer facing adversity."
Source : PA
Source: PA