Healing in the House of Elrond

Frodo was healed from his Morgul wound thanks to the healing from Elrond. The poison was removed through removal of the splinter & through using healing techniques such as Athelas. However, even still Frodo in theory should have died from the severe infection caused by the wound, given that it stayed in him for about 2 days before Elrond removed it. So how was Frodo saved? Was it through Elrond using healing powers inherited from Eru?