Alcaraz, also the reigning Queen's champion, lost 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 Spaniard was undone by Draper, 22, 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. Carlos is the defending champion, he won Wimbledon, he's an incredible talent and amazing for the sport," said 31st-ranked Draper, who won his first ATP title in Stuttgart last week.
"I had to come out and play well and luckily I did. There's no place I'd rather be right now."
Alcaraz had not been beaten in seven weeks or indeed lost a match on grass in nearly two years.
But that record was of little concern to Draper as he became the first British man to beat a top-two player on grass since Murray defeated Novak Djokovic in the 2013 Wimbledon final.
An enthralling first set ended in a tie-break after neither Alcaraz nor Draper had managed a break point.
But it was Draper who pulled clear with several fine serves to earn five set points, with the Briton taking the third when Alcaraz hit a return into the net.
Left-hander Draper earned the first break point of the match at 3-2 up in the second set and duly converted it with an excellent return of serve down the line.
Alcaraz saved three match points on his own serve before Draper, a day after 37-year-old Murray limped out injured of Queen's after just five games, finished the job for a win that appeared to signal a changing of the guard in British tennis.
Draper will next play American fifth-seed Tommy Paul, a 6-3, 6-4 winner over Chile's Alejandro Tabilo, in the quarter-finals.
