I actually had to research this one. Of all the answers I found this one explained what I had originally thought, here ya go!
Dumbledore was the master of the Elder Wand.
Wands can only transfer wizards if they win allegiance over it. Allegiance can be won by overpowering the other wizard, usually by death, but not always.
Before Dumbledore was killed, Draco Malfoy disarmed him. By doing so, it disallowed Dumbledore to fight back. When Snape cast the killing curse, Dumbledore was without a wand and had no means to defend himself.
Therefore, Draco was the true master of the Elder Wand because although Snape physically killed Dumbledore, Draco made it impossible for Dumbledore to fight back. He overpowered the wizard, and became the true master of the Elder Wand.
Of course he was unaware of this. Now in Malfoy Manor, Harry and Ron steal some wands to fight back for Hermione, but Bellatrix makes them drop them or else she would kill Hermione. Draco was ordered to pick them up and he placed the wands in the same hand as his own wand.
When Dobby distracted all the wizards, Harry had a mini-wrestle with Draco and took all the wands back, including Draco’s own wand. He overpowered Draco by force. Even though Draco’s wand was NOT the Elder Wand (it was still with Dumbledore), the allegiance of the Elder Wand transferred to Harry. Note: there is no need to touch or physically take the Elder Wand in order to be its owner. All you have to do is overpower the wizard that owns the Elder Wand.
So now Harry is the owner of the Elder Wand. And he just took Draco’s wand. Draco’s wand is now his wand. Voldemort took the Elder Wand from Dumbledore, thinking it would automatically transfer to him, but because he never won allegiance, it did not. He didn’t know this.
When Voldemort (holding the Elder Wand) cast the killing curse at Harry, the Elder Wand refused to kill its own master (Harry). The spell rebounded and hit Voldemort, killing him. The Elder Wand recognized Harry and recognized Draco’s wand and its true master. Remember, wands will not kill or harm its master.
