The Red Devils started the game brightly but fell behind on 36 minutes when defender Victor Lindelof diverted the ball past his own goalkeeper.
David De Gea failed to deal with Leighton Baines' corner in from the right, and the ball hit Lindelof at the far post and trickled over the line.
Manchester United hit back in the second half through their 18-year-old substitute. The teenager fired a crisp shot that beat Jordan Pickford at his near post to secure a point.
The Red Devils made history Sunday with their team selection, naming a squad containing an academy player for the 4000th game in a row:
Manager Ole Gunnar Solskjaer stuck with the same team that beat Manchester City last time out in the Premier League, while Everton switched to a back five and brought in Seamus Coleman, Yerry Mina, Tom Davies and Bernard.
Goal shared the two teams:
The hosts started quickly and fashioned a chance in the first minute. Anthony Martial and Fred combined to tee up Jesse Lingard inside the penalty area, but he could only turn and put a shot wide of the Everton goal.
Daniel James and Marcus Rashford also had sights of goal in a bright opening but neither player could find the target. Rashford also forced Pickford into a strong one-handed save from his powerful free-kick from just outside the penalty area.
Everton grew into the game as the half progressed and took the lead before half-time. De Gea could not punch a corner clear under pressure from Dominic Calvert-Lewin, and the ball hit Lindelof and bounced over the line:
De Gea did appear to have been fouled by Calvert-Lewin, but the goal was given after a VAR check which drew some surprise:
Manchester United struggled to muster a response before the break, lacking the invention to break down a committed and disciplined Everton side who made it to half-time with their lead safely intact.
The Red Devils saw plenty of the ball in the second half but struggled to create clear chances initially.
Luke Shaw called Pickford into action with a fierce shot that the goalkeeper parried. The ball fell kindly to Daniel James, but he could only hit team-mate Lingard full in the face with his follow-up.
Solskjaer sent on Mason Greenwood for Lingard for the final 25 minutes of normal time, while Everton also made an attacking change and brought on Moise Kean.
The substitution was to prove crucial for the hosts as they finally found an equaliser on 78 minutes. Greenwood fired a shot that flew through Mina's legs and squeezed inside Pickford's near post:
The Red Devils went in search of a winner but could not find another goal, while at the other end, De Gea needed to be alert to tip away a swerving shot from Alex Iwobi.
There was also late drama when Everton replaced substitute Kean with only minutes of normal time remaining.
ESPN FC's Mark Ogden noted the 19-year-old's reaction to the change:
The draw sees Manchester United move into sixth place in the table and now just four points behind Chelsea in fourth place, while Everton are unbeaten in two games under caretaker manager Duncan Ferguson.
Both teams are back in action on Wednesday in the quarter-finals of the Carabao Cup. Manchester United host League Two side Colchester United, while Everton play Leicester City at Goodison Park.