Draco never "got" the wand, per se. However, he became the elder wand's master by disarming its previous master, Albus Dumbledore, at the top of the Astronomy Tower the night Dumbledore was killed.
It is stated in the books that one does not have to kill a wand's owner to take mastery of his or her wand (although this often happens). One must only properly defeat the witch or wizard. Because Draco legitimately disarmed Dumbledore, and Dumbledore never reclaimed the wand, Draco became the master of the wand.
Voldemort incorrectly inferred that because Snape killed Dumbledore that Snape must have disarmed Dumbledore and become the master of the wand.