This seems to be something that people aren't sure about. So far, I've found two sites that both mention that when Delta was resurrected, it was incorporated into him:
Maybe when Delta was ressurected, Elenor incorporated it into his structure.
Tenenbaum helped Eleanor to revive Delta, i am sure she sneaked the saving ability into there at the same time.
One thing that I've noticed on the Wiki for the Alpha Series Big Daddys is that all Alpha Series began their life as a test subject experimenting with Plasmids. Because of this, they still retain some of the abilities:
Each member of the Alpha Series began as a test subject for various Fontaine Futuristics Plasmids, and they retain some Plasmid abilities as a result.
I would think that maybe Subject Delta was part of an experiment that included the ability to purge ADAM Sickness, and because of this, he still has the ability to do so.