Goals from Wilfried Zaha, Christian Benteke, Luka Milivojevic and Patrick van Aanholt secured Palace's own safety and sent Hull down.
Hull could not afford to lose after Swansea had opened a four-point gap with a 2-0 win at Sunderland.
Swansea's survival marks a remarkable turnaround in fortunes under Paul Clement.
Clement, Swansea's third manager of the season after Bob Bradley and Franceso Guidolin, took over a side in January that was bottom with only 12 points from 19 game.
The former Derby County manager's arrival at the Liberty Stadium saw an initial revival with Swansea winning three of his first five games in charge.
But between the beginning of March and mid-April the Swans claimed only a single point in six Premier League games.
Swansea's 2-0 home win over Stoke City on 22 April was the start of a three-match unbeaten run which boosted their hopes of escaping relegation.
Their 1-0 home victory over Everton saw them leapfrog Marco Silva's Hull, who had lost 2-0 at home to Sunderland earlier on Saturday, 7 May.
And Swansea extended their unbeaten run to four games with a win in their penultimate game at already relegated Sunderland, which opened a four-point gap between the Welsh side and 18th-placed Hull.
Hull's defeat at Selhurst Park sealed the Tigers' fate and ensuring Swansea will remain in the Premier League for a seventh successive season.
Swansea will play in the Premier League next season after Hull were relegated after a 4-0 defeat at Crystal Palace.