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The world number six briefly threw away a final set advantage - being broken at 5-4 when serving for the match - but he responded brilliantly to secure a 6-3 5-7 7-5 victory in Group Pete Sampras.
But Thiem hit back, halting the Bulgarian as he served for the match - only to then immediately donate his serve with two untimely double faults.
Into the tie-break, where two crisp volleys sent Murray and Soares 4-1 ahead and they took the second of four set points when Mike Bryan returned long.
Dimitrov, 26, just about held his nerve to win a gripping battle of the one-handed backhands against Thiem, who is playing in the Finals for the second year running.
Dimitrov will play Nadal on Wednesday if the Spaniard beats David Goffin later today, and he will be full of confidence after producing some great net play and battling through after Thiem made his second set comeback.
"I'm not going to lie - I was pretty nervous, my first match out here", said Dimitrov.
Thiem applied pressure in game eight but it came to nothing, however with Dimitrov serving for the match for a second time the Austrian finally found a way through to level the decider at 5-5.