Adrian came close to adding a third before the break, but he scuffed his shot, smashing the ball into the side-netting.
The hosts started to get more possession towards the end of the first half and continued their momentum after the break, but Turan was able to add his second, turning inside the box to slide a low effort into the bottom corner.
Diego Costa came off the bench for the final 15 minutes and almost had an immediate impact, firing just wide from close range less than two minutes after coming on.
The home side fought hard late on as they tried to keep the tie alive, but David Villa was able to make it four following an end-to-end spell, striking it on the turn from close range.
The second leg of the tie takes place later this month.