The 27-year-old Scot will now have a last eight clash with Australian teenager Nick Kyrgios.
The match was a high quality affair with brilliant stroke play, lengthy rallies and comebacks from both players throughout.
When 5-2 down in the fourth set Murray saved a set point while serving at 5-2 down, then pulled back to 5-5, and then broke Dimitrov to serve out the match for the win.
"I think I got quite lucky at the end, a few net cords went my way, that was the difference really," Murray told reporters.
"The momentum was switching both ways all the time," he added.