Uruguay international Edinson Cavani headed home a Rashford cross to get the Red Devils up and running before Portugual international Bruno Fernandes scored a brilliant shot in first-half added time to put Manchester United in the control.
However, the Red Devils also lost France international Paul Pogba to a thigh issue - and the Tofees were level within 7 minutes of the 2nd half as Ole Gunnar Solskjaer's backline imploded.
Spain International David de Gea spilled a tame Calvert-Lewin cross into the path of Doucoure to score from close range after forty nine minutes.
Doucure then turned provided 3 minutes later as he teed up Columbia Star James Rodriguez to put home a low half-volley, with Manchester United's backline slow to react to the danger.
With the game finely balanced, English left-back' Luke Shaw whipped in a fine free-kick for Scotland Star McTominay to head home with twenty minutes left.
However, the Tofees won a free-kick in the 4th minute of added time, and English Striker Calvert-Lewin poked a close-range volley home to win a point Everton.
The result leaves Manchester United 2 points off leaders Manchester City, who have 2 matches in hand.
This is how the table looks like
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