Draper was dominant throughout the match against Murray as the young Briton held off a set point in the first set before bouncing back to win the second comfortably against his compatriot.
Despite the impressive victory, Draper explained post-match that he still looks up to and admires Murray for all that he has achieved in the sport.
"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 comes after improving form for the 21-year-old, who also knocked out Dan Evans and Leandro Riedi, in the previous rounds at Indian Wells.
But the youngster is not becoming complacent and said he will continue to work on different parts of his game to improve further moving forward.
"I want to keep on trying to impose my game and use my weapons. It's not easy in these conditions. It's quite slow tonight, the wind was a bit swirly and obviously playing against Andy, he makes it incredibly awkward for you. It's something I'm looking to improve with my coach and it's definitely starting to pay off a little bit more."
In the round of 16, Draper will face Spanish star and US Open champion Carlos Alcaraz in what will prove to be a stern test for the developing Brit. Draper explained that facing Alcaraz will be tough, but it is something he is looking forward to.
"Carlos is another level up, he's been No 1 in the world, had an exceptional year last year, so it would be amazing to play him again in another real battle."