Draper won 7-6 (6) 6-2 in two hours in his first ATP Tour clash with his fellow Briton.
Murray had a set point in the first but Draper saved it with an ace and the 21-year-old went on to book a fourth-round clash with top seed Carlos Alcaraz.
In an on-court interview broadcast by Amazon Prime Video, Draper said: "I've looked up to Andy since I was so young.
"I watched him win Wimbledon for the first time in 2013 and then I've had the opportunity to get to know him and practise with him often since 2019.
"He's a really special person, a great champion, great human being and I'm privileged to play against him on this court."
Draper's win has lifted him up to 43rd in the live ATP rankings, having started the week at 56th, one place below the 35-year-old Scot.
Draper, who defeated British number two Dan Evans in the second round, added: "Beating Dan and Andy, that's about as much confidence as I'd need for that (Alcaraz) match.
"Carlos is another level up, he's been number one in the world, had an exceptional year last year, so it would be amazing to play him again in another real battle."
Alcaraz defeated Tallon Griekspoor in the third round to become the second fastest player to reach 100 ATP Tour wins.
The 19-year-old now has a 100-32 win-loss tour-level record and is behind only John McEnroe (100-31) on the list of players to reach the milestone, ahead of Andre Agassi (100-35), Rafael Nadal (100-37), Mats Wilander and Jimmy Connors (both 100-38).