The January signing from Besiktas struck from inside the box either side of Eric Maxim Choupo-Moting's leveller in a dramatic conclusion to the match.
A red card shown to Charlie Adam 30 minutes in had been the only real drama up until the frenetic finale, as the Toffees held on for just a second away win of the season.
Relive how the 90 minutes of action unfolded with Sports Mole's live text coverage below.
STARTING XI: Butland; Johnson, Zouma, Shawcross, Stafylidis; Allen, Adam, Badou; Shaqiri, Crouch, Ramadan
SUBS: Haugaard, Bauer, Berahino, Choupo-Moting, Jese, Martins Indi, Fletcher
STARTING XI: Pickford; Baines, Keane, Jagielka, Coleman; Davies, Gueye, Rooney; Bolasie, Tosun, Walcott
SUBS: Joel, Schneiderlin, Martina, Niasse, Klaassen, Calvert-Lewin, Holgate
Paul Lambert: "I know what's above me, I know the points. I know everything that is going on but I concentrate on my own team. I have always been like that. I have never tended to look at other people's results. I've always thought 'look after yourself first and foremost'. The main objective is stay in the league and then rebuild for next season."
Sam Allardyce: "We've tried a number of things. Changing system, changing personnel - but changing mentality is the big one. We've been using some more work into the mind in the analysis room. Who we are, what we want to achieve, what we can get. Visualising the performance you give at the top level and thinking of that to put yourself in a positive frame of mind."
STOKE CITY SUBS: Haugaard, Bauer, Berahino, Choupo-Moting, Jese, Martins Indi, Fletcher
EVERTON SUBS: Joel, Schneiderlin, Martina, Niasse, Klaassen, Calvert-Lewin, Holgate