Alcaraz, the defending Cinch champion, then struck two backhand errors to face set point.
Alcaraz saved two set points before netting a down the line backhand return to drop the tiebreaker.
In set two the 6 feet four inches tall Draper continued to mix powerful serves and aggressive groundstrokes and net play to take a 2-1 lead.
Then with Alcaraz serving down 2-3 30 all, Draper struck a down the line forehand volley winner to reach 30-40 and the first breakpoint of the match.
Draper took the following backhand return early; striking an aggressive down the line backhand return winner to break serve and lead 4-2.
Draper continued to use a big lefty serve and net rushing game to fight off a break point in the following game and to hold serve to lead 5-2.
With the 6 feet tall Alcaraz serving to stay in the match, Draper struck several aggressive crosscourt topspin forehands to reach match point at 15-40.
Alcaraz then stayed three match points in a six deuce game to hold serve and reach 3-5.
Serving for the match, Draper committed two backhand errors to trail 0-30.
Alcaraz then netted a backhand return before also netting a down the line inside out forehand return to reach 30 all.
Draper struck an ace down the tee to reach his fourth match point of the day.
Alcaraz netted an inside out crosscourt forehand to end the match after one hour and 39 minutes of play.
Draper, who struck eight aces in the match, jumped into the air in a celebration after the match.
Speaking to the crowd following the match, Draper said "you guys are amazing. I want to keep on playing here. I've been desperate to come back."
Draper did not lose serve for the day in defeating the 2023 Wimbledon champion Alcaraz.
In Friday's quarterfinals, Draper takes on the fifth seeded American Tommy Paul.
Following his success this week, Draper is set to move from 31 to at least 28 in the rankings and is most likely guaranteed one of the 32 seeds at Wimbledon which begins July 1st.