Paul Lambert’s side are second bottom, three points from safety, with only one game remaining.
With fourth-bottom Swansea and third-bottom Southampton playing each other in midweek, it is impossible for Stoke to collect enough points to avoid slipping into the Championship.
Stoke have spent 10 seasons in the Premier League, but a run of 13 games without a victory ensured that streak will come to an end.
Stoke goalkeeper Jack Butland was in tears at the final whistle and City fans were just as emotional as the realisation of their team’s relegation sank in.
Hundreds left the bet365 stadium in the closing minutes before those that remained sang “City til I die” in a defiant rallying cry.
The blame for Stoke’s predicament lies jointly with Lambert and Mark Hughes, who was sacked in January after a wretched run culminated with an embarrassing FA Cup third round defeat against fourth tier Coventry.
The Potters won their first match with Lambert in charge against Huddersfield, but that was a false dawn as they remained deep in trouble for the rest of the season.
Hughes could suffer the indignity of having a role in two relegations in one season as his Southampton side, who face Everton later on Saturday, are in severe trouble themselves.
Xherdan Shaqiri had described the Palace clash as “a war we have to win” and the Stoke star landed the first blow in the 43rd minute.
Having been fouled on the edge of the penalty area, Shaqiri picked himself up and exacted his revenge.
Shaqiri’s superb left-footed free-kick took a slight deflection off Palace midfielder Ruben Loftus-Cheek as it curled past Wayne Hennessey before he could react.
It was the Swiss forward’s first goal since February and he celebrated by sprinting to the touchline to jump into Lambert’s arms.
But Stoke’s nerves were still frayed.
They couldn’t cling onto their lead and Palace midfielder James McArthur equalised with a clinical finish in the 68th minute.
Forced to throw men forward in search of a winner, Lambert’s side were carved open in the 86th minute as van Aanholt slotted home to seal Stoke’s fate.
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