"My impression of the luck elixir that if you were to get an item, say a purple (epic) item, it would have a chance to re-roll and bump your item to the next level of rarity, (legendary in this case)."

You are re-rolling from one loot table to another loot table. Loot tables can contain from 1 to however many items the designer assigns to it. So, for example, if you are in the lowest loot table (with its list of items) and get a re-roll on a common, it's entirely possible that the next higher loot table (with its own list of items) also contains some commons that you have a chance at rolling to, as I understand it. Mostly, you'll get a better item but sometimes not.

Hope this helps.