Stoke led early through Steven Nzonzi but Clint Dempsey -- who set up Adebayor's winner -- equalized before the home side had Charlie Adam sent off just after halftime.
Arsenal can regain fourth place with victory over Wigan on Tuesday in a match that the third-bottom team needs to win to have any chance of avoiding relegation.
Moyes will leave Everton in sixth place following a 2-0 win over West Ham, with both goals coming from Belgian striker Kevin Mirallas.
Liverpool stayed seventh with a 3-1 win at Fulham as a hat-trick from striker Dean Sturridge meant the London side may need a point in the final round to avoid relegation.
Newcastle and Norwich both won on Sunday to ensure another season in the top flight, respectively beating bottom club Queens Park Rangers 2-1 and eighth-placed West Brom 4-0.
Sunderland could have condemned Wigan to the drop just a day after the club's biggest success at Wembley, but drew 1-1 with Southampton to stay in the relegation mix.