Doku was a constant threat for Pep Guardiola's side, who led, trailed and fought back to earn a point at the Hill Dickinson Stadium.
The Belgian opened the scoring with a wonderful curling effort to put City ahead and repeated the trick in second-half stoppage time to snatch a draw.
He also completed five dribbles, won 14 duels and drew seven fouls as Everton's players struggled to cope with his pace and trickery.
Doku's display, though, which earned him a 9.6 Flashscore player rating, will be slightly overshadowed by his team losing ground to Arsenal in the title race.
City will look to bounce back when they host Brentford on Saturday evening.