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Kevin-Prince Boateng came close for the hosts early on as he hit a long-range effort, which flew just wide of the left post.
Max Meyer was wasteful for Schalke as he found himself with plenty of space inside the box, yet he somehow managed to drag the ball wide of the goal.
The home side finally made Braunschweig pay in the 17th minute when Meyer found Goretzka outside the box, leaving him to drill an effort into the bottom left corner of the goal.
The hosts went further ahead in the 65th minute through a standout individual effort from Huntelaar, who smashed home a free kick into the centre of goal.
Schalke put any chance of a Braunschweig revival to bed when Szalai lifted the ball into the top corner deep into injury time.