Roberto Martinez's side were under immense pressure at times but were able to hold out City's attack to come away with a well-earned point.
Toffees goalkeeper Tim Howard put in a commanding display, keeping out a Yaya Toure header in the first half, before twice denying Sergio Aguero in the second half among other pieces of defensive brilliance.
Leon Osman's sumptuous volley from Romelu Lukaku's lofted ball was only inches wide for Everton who could also count themselves lucky not to concede a penalty when John Stones slid in on Raheem Sterling at the death, taking the legs of the winger, but not the ball.
All in all, Everton managed to frustrate Manuel Pellegrini's City who remained third on the table, three points in arrears of leaders Arsenal and Leicester City, as the Toffees stayed 11th following a third straight winless league fixture.
2015/16 BARCLAYS PREMIER LEAGUE - SEASON SCORES