Sinner, the No3 seed who had reached the fourth round of the preceding French Open, had looked to be cruising at 4-0 in the opening set.
But a previously nervy Draper settled into the match to force an opening tiebreak. Again, he fell a break behind in the second set but broke Sinner late on to force another tiebreak for a 7-6, 7-6 victory.
Afterwards, Draper, the world No309 said: “I thought I had no chance of beating Jannik today. I got off to a slow start. Once I won a couple of games, I progressed from there.
“I’m just glad I was able to play well against such a great player like Jannik. He’s top 20 in the world, way ahead of me. It’s great to beat him on this court.”
It proved two British wins out of two on Centre Court as British No2 Cameron Norrie recovered from a set down to defeat Spain’s Albert Ramos-Vinolas 4-6 6-3 6-4.
Norrie had never won in four previous attempts at Queen’s and said: “It’s so nice to get my first win here. I’m just really enjoying my tennis. In big moments, that showed today. The thing I’m doing is the fundamentals pretty well. “
The last Briton in singles action, Liam Broady, could not make it a clean sweep of home wins, though, as he lost 6-4 6-4 to Adrian Mannarino.