The British duo will now face Spanish pair Tommy Robredo and Maria Jose Martinez Sanchez in Saturday’s final.
In beating Russia, Murray and Robson have become the first British team to reach the final of the competition.
The pair beat Igor Andreev and Elena Dementieva 6-4 6-7 7-6 to finish top of Group B and maintain their perfect record in the competition.
“I have to say thanks to Laura because she played great,” said Murray after their victory.
“In the mixed, the guy is supposed to try to dominate and help the girl but every time I tried that we lost the point, so I left it to her after that.”
Elsewhere Rafael Nadal and Roger Federer are still on course to meet in the final of the Qatar Open after both made it through to the semi finals of the event.
Federer overcame Ernests Gulbis in three sets and Nadal was 6-1 2-0 up in his quarter-final when Steve Darcis was forced the retire through injury.
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BIOGRAPHY: Marcus Rashford