Third seeds Peya and Soares, of Austria and Brazil respectively, claimed their seventh ATP World Tour doubles title with the win over the Brits.
Briton Murray lost his serve early on, as the favourites took a 3-2 lead in the first set, as Peya and Soares held their nerve to take the opener.
Murray’s partner Fleming then failed to hold his serve in the very first game of the second set. The British pair fought hard and broke back, however, levelling at 4-4. A set point came and went as the second set entered a tie-break.
Fleming double faulted at 2-5 before Peya and Soares sealed the victory.
"We played a solid match overall," said Peya. "We struggled a little bit in the middle of the second set and when we got close to the end. But I think we hung in there well and played overall a good match."
Soares also chipped in to express his delight at beating the two Brits: "It feels very good.
"This was our second final in a Masters event. It was tough conditions, tough opponents. We played a solid match and we're very happy with the win."