What happens when I control my opponent and they draw from an empty library.

So, i was playing my jank Aetherworks Marvel deck and came across a strange rules interaction. I had managed to mill out an opponent with a couple of Ulamogs (100 card lifegain deck; I guess they just really didnt want to loose to mill). They had ragequit by that time and i had a free turn to do whatever i wanted and I spun Marvel one last time and turned up a Emrakul. Unfortunately, it was the last win that i needed for the night so i didnt go for it, but i was worried if I did, I might loose the game by drawing from an empty library. I don't think that is the way it works, but i don't know for sure.

So, my question is, what happens when i control my opponent when they attempt to draw from an empty library? I think that since they are still the ones drawing and I am the one controlling them, they will loose but, again, not sure. All i can find when googling is an AI answer which i don't trust.