Draisaitl won it after Connor McDavid's cross-crease pass deflected off Sharks forward Joe Pavelski's stick, hit Draisaitl's skate and went inside the left post.
Hitchcock replaced Todd McLellan, who was fired on Tuesday.
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"I was more worried that it was past my bedtime and I wasn't going to be able to stay up," Hitchcock said. "I don't think I've seen a third period all year out West. But for me, we just got better and better as the game went on. We got better in the second, we were really good in the third.
"We really got on the grind. There's a lot of weight and size in this group, a lot more than I can remember, and we started to lean pretty hard in the third period and used it, and it was very, very effective. It was a great win."