The newly-promoted world number two was broken by his Bulgarian opponent in the very first game of a three-hour encounter and only just managed to drag 16th seed Dimitrov back from a double-break lead to draw level in the eighth game of the second set.
Murray then forced a decider by comfortably winning a tie-break but the tide would turn against the Scot once again.
The 2013 Wimbledon champion was a break up after two games, yet Dimitrov rallied to win five straight games, wresting back the initiative before a crucial, fascinating 32nd game.
The Bulgarian looked more than ready to break Murray and seal his progress, enjoying two match points during a frantic display of rallying and cross-court counter-punching, but it was not to be.
Murray somehow held his serve, broke Dimitrov, and held once more to clinch a 4-6 7-6 (7/3) 7-5 victory.
A sixth career win over Dimitrov means Murray, whose triumph at the Rogers Cup saw him leapfrog Roger Federer in the battle to replace Novak Djokovic at the top of the ATP Rankings, gets the opportunity to face French number two Richard Gasquet in the last eight at the Lindner Family Tennis Center.
That engagement will come around rather quickly for Murray, as it is scheduled to commence at 2.30pm local time on Friday.
But the Scot, who will tackle either Federer or Spain’s Feliciano Lopez if he reaches the semi-finals, insists he feels ready to challenge all comers.
“If you want to win the big competitions, you’ll have to come through difficult situations,” Murray told atpworldtour.com. “Last year, I didn’t come through enough of them.
“I lost a lot of matches from winning positions, and this year I managed to turn that around, which was very positive.”
Dimitrov, who let a 5-2 lead slip away in the dramatic final set, added: “I was happy with the way I played at least, but that’s about it.
“I had too many opportunities. That’s the only thing. If that you have many and you don’t win, that’s it.
“Everything else felt pretty good. The body held up well for three hours, so good things ahead.”