Jerome has been allowed to leave the Britannia Stadium after Stoke brought in Marko Arnautovic from Werder Bremen earlier in the day.
He scored 12 goals in 60 appearances during his two seasons at Stoke, but had not found favour during Mark Hughes' early weeks as manager.
The 26-year-old's sole appearance for Stoke this season coincidentally came as a late substitute during the 2-1 win over Palace just over a week ago.
Jerome was the fourth player that promoted Palace signed on deadline day as he followed Jack Hunt, Adrian Mariappa and Barry Bannan to Selhurst Park.
Palace boss Ian Holloway had also been trying to bring Queens Park Rangers striker Andrew Johnson back to the club and held talks over a deal for Arsenal forward Nicklas Bendtner.