But City's lead didn't last long as within 6 minutes the Toffeemen were ahead themselves.
On 62 minutes, Scot Gemmill chipped Schmeichel from the edge of the area after dispossessing Paul Murphy.
The Irish defender was again caught out 4 minutes later when he handled outside the box and Schmeichel couldn't prevent Duncan Ferguson's low free-kick from going in.
City were gifted a chance to level on 78 minutes when Turner brought don Lee Croft and Chris Shuker made no mistake from the spot.
And with just 8 minutes remaining, Christian Negouai snatched the decisive goal. The ball broke free after a tussle between Jamie Tandy and O'Hanlon before Karl Bermingham's shot was saved but the French midfielder was on hand to convert the rebound.
City: Schmeichel; Murphy, Jordan, N D'Laryea, Onouha; Negouai, Bosvelt (J D'Laryea 65), Sibierski (Bermingham 65), Berkovic (Tandy 65); Croft, Shuker.
Subs: Matthewson (gk), Warrender.
Everton: Turner, O'Hanlon, Wilson, Clarke, Nyarko, Osman, Watson, Gemmill, Campbell (A Moogan 72), Ferguson, Alexandersson.
Subs: Symes (gk), A Gerrard, Wynne, Hughes.
Attendance: 516.