Carlo Ancelotti's side were forced into extra time, and produced just enough quality when it mattered to see off their dogged opposition.
But the home side weren't impressive for large swathes of the game.
But who were the standout performers, if any, in the contest?
❌ BAD PERFORMANCE
Alex Iwobi: 5.7
The former Arsenal player produced an horrendous performance for Everton.
Iwobi was listless for large portions of the match, constantly giving the ball away and didn't seem in the mood for the challenge.
He was hauled off in the second half, to no one's surprise.
Tom Davies: 5.7
Davies, like Iwobi, produced a performance that won't live long in the memory.
Played in central midfield, Davies was outfought at times, and repeatedly give the ball away to the opposition players.
The midfielder hasn't started too many games under Ancelotti, and his performance wouldn't have won him any favours with the Italian coach.
⭐ STAR MAN
Abdoulaye Doucoure: 7.6
Doucoure came off the bench to save Everton's blushes.
The midfielder was everything Davies and Iwobi weren't: committed, interested, and able to produce quality when it mattered.
Doucoure's goal was a smartly-taken effort, curling the ball from the edge of the area and into the bottom corner of the Rotherham goal to send The Toffees into the fourth round.