Arsene Wenger says that his Arsenal team must stay strong as they seek to overhaul Liverpool or Manchester City in the race for the top four.
Tottenham (2nd with 77 points) is guaranteed of not finishing lower than 3rd, that is in the event they lose their last two games.
Sanchez and Arsenal are next in action against already-relegated Sunderland on Tuesday.
"It's a huge result for us", said Swansea manager Paul Clement, whose side host West Brom on the season's final day.
While they're not quite there yet - much will depend on the extent of Alexis' injury - you can be sure that they will do everything possible to win their final games and hope that Liverpool slip up to give them a golden opportunity of retaining their seat at Europe's top table.
Stoke's reply came through substitute Peter Crouch in the 67th, which made it 2-1.
Sanchez's goal was his 50th in the Premier League, his 21st of the season and his 15th away from home. I will do that as long as I am here, with full commitment. "I was surprised he got to the ball when I saw his position, and people told me it was handball".
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The impressive win must have been particularly sweet for Wenger, who has suffered a poor record on his visits to Stoke and had not won in his previous eight away league games against Hughes's teams.
Asked if it had been unfairly rough, Wenger said: "I don't say that". That's Alexis, he's a fighter who always finds resources.
Sanchez subsequently came off, apparently with an injury that Arsenal will certainly hope is only minor.
"We were okay and had a good shape in the first half, but in terms of what we wanted to do in attack, that didn't really happen for us".
Hughes said: "Them scoring just before half-time was obviously a blow to us and then, conceding the second goal, it is hard from that point onwards".
"We had good moments when we had numbers to break, and it just kept on breaking down - and we're better than that".
The game quickly descended into a scrappy affair but Sanchez reasserted Arsenal's dominance with a cross-goal finish from the edge of the box, a touch off Shawcross helping it on its way.
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