Spaniard Alcaraz, also the reigning Queen's champion, fell 7-6 (7/3), 6-3 with the British number one claiming the biggest win of his career.
World number two Alcaraz arrived for this grass-court warm-up event for Wimbledon fresh from his French Open triumph on the clay of Roland Garros.
But the 21-year-old was undone by Draper, who is bidding to become the first British men's singles champion at Queen's since Andy Murray won his fifth title at the event in 2016.
"It was a really tough match I knew coming out here that Carlos is the defending champion, he won Wimbledon, he's an amazing talent," said 31st-ranked Draper, who won his first ATP title in Stuttgart last week.
"I knew I had to come out here and play really well, and luckily I did that."
