Ayew struck with a vicious swerving effort from 20 yards to give Swansea just an eighth point from 10 home Premier League games this season, leaving them four points adrift of safety.
Under the guidance of temporary player-manager Leon Britton after Paul Clement was sacked, the hosts seemed energised but failed to trouble their opponents in a goalless first half.
Palace posed a threat on the break and, after Ruben Loftus-Cheek was brought down by Federico Fernandez, Luka Milivojevic stroked his penalty down the middle to give Roy Hodgson's men the lead.
But Ayew's goal gave Swansea a point as their search for a new manager continues, with former Palace manager Frank de Boer and ex-West Ham boss Slaven Bilic among those under consideration.
For Palace, the draw extended their club record unbeaten run in the Premier League to eight games but they slipped to 16th in the table.