The Glaswegian moved to a career high of 200 on the ATP Rankings on Sunday and his emphatic 6-1, 6-4 victory over American Young, who beat Andy Murray last year, will now take him even higher.
Baker broke the Amer-ican five times and notched 11 break points during a match that lasted less than an hour.
He will now face either seventh-seeded German Philipp Kohlschreiber or Go Soeda of Japan, who are due to meet today
Baker said: "This win means the world to me. I knew if I could maintain the level I produced against (Jo Wilfried) Tsonga last week at Queen's I could trouble Young and I did.
"I was aggressive when I had to be and really tried to take the game to Young. In terms of the ATP Rankings, this was my best win. That is a no-brainer.
"The great thing for me is the amount of belief I can take from this win. I am 200 in the world and my next goal is to break the 150.
"But now I need to back this win up against either Kohlschreiber or Soeda. But I will walk on to the court against either of them full of confidence and that is a massive plus."
Baker added: "To have been given wildcards at both Queen's and Eastbourne, and now made the most of these by getting wins in both tournaments is huge for me. It is great to make the most of these opportunities."