Stanway had earlier been denied what looked a certain goal by Megan Finnigan's superb tackle, with Everton goalkeeper Sandy MacIver also pulling off a string of outstanding saves while Caroline Weir twice hit the woodwork.
City had hit the front through Sam Mewis' 40th-minute header, with Valerie Gauvin striking back for the Toffees, who were aiming to win the FA Cup for the first time in a decade.
Despite a stern rearguard effort, however, Everton's resilience was broken when substitutes Stanway and Janine Beckie secured City's third FA Cup triumph in four seasons.