Women's Euros: Stanway screamer sends England into semi-finals
The Lionesses had not lost in 17 games under coach Sarina Wiegman but fell behind to Esther Gonzalez's second-half finish before substitute Ella Toone equalised with six minutes to go.
Stanway smashed in the decisive goal in the 96th minute and England will next face either Sweden or Belgium, who meet in Leigh on Friday, after making the last four at a major tournament for the fourth successive time.
Ellen White thought she had opened the scoring in a cagey first half but was denied by the offside flag, before Gonzalez finished into the bottom-left corner after Athenea del Castillo's cutback in the 54th minute.
Mary Earps kept England in the game with a smart save from Del Castillo's deflected cross and the Lionesses capitalised as Toone volleyed home from Alessia Russo's cushioned header to send the match to extra time.
Stanway then delivered the decisive moment, receiving the ball from Keira Walsh before driving forward and arrowing into the top-left corner to keep England's Euro 2022 hopes alive.