if you are in a game where azmodan quest is open, but you have the chance to turn in the quest to get the reward, you can leave that game, just click resume when u get into the menu, and turn the quest in there instead.
If you do it this way, the first game u got the quest in, wont be affected, so you can join that game again and turn the quest in there aswell.
Thats how this entire thing work, u turn in quests in other players game, + your own, for the maximum amount of 5 for 1 char.
Im currently testing a boost system now, i started Azmodan normal at level 1
Nightmare at level 25
Hell at 50
Current playtime 2 hours 59 minutes and im level 52

With perfect turnins ( 5 x turnins at hell diff) you can go from level 59 to levle 60 in 2 runs + a few mobs