Blackpool, who last saw top-flight action in 1971, secured a place in the play-offs alongside Nottingham Forest, Cardiff City and Leicester City.
Wednesday needed to win their final game against Palace to survive - and send the Londoners down.
They were trailing 2-1 before a Darren Purse equaliser after 87 minutes set up an amazing finale.
Wednesday bombarded the Palace penalty box from all angles through five minutes of stoppage time only for the visitors to hold out and stay up.
Third-placed Forest were already assured of a play-off place and will face Blackpool next weekend after the Lancashire side came from behind to draw 1-1 at home with Bristol City.
That would not have been enough if Swansea had beaten Doncaster, but the Welsh side could only draw 0-0.
Cardiff and Leicester will meet in the other playoff semi-final with the final at Wembley on May 22.
West Bromwich Albion and champions Newcastle United had already secured the automatic promotion places and Newcastle signed off with a 1-0 win at Queens Park Rangers which took them to 102 points.
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