Murray had beaten old rival Rafael Nadal on Friday to book a showdown against world number five Rublev but the 2009 and 2015 champion lost the first set before slipping 4-2 down in the second.
The Scot won three games in a row and although he was unable to force a decider as Rublev claimed a 6-4 7-6 (2) victory, Murray can reflect on an encouraging week having beaten British compatriot Dan Evans and then Nadal.
The first four games on Saturday evening went with serve but Murray then came under pressure from his opponent. A first break point was saved with an ace but Murray miscued on the second as Rublev went up 3-2.
It was all Rublev, who at 24 is 10 years younger than three-time grand slam winner Murray, needed as he went on to take a one-set lead, continuing to punish his foe with powerful groundstrokes from both wings.
Murray's serve had helped to keep the scoreline competitive but he was broken again at the start of the second set and the writing looked to be on the wall as he continued to be picked apart by Rublev.
However, Rublev inexplicably allowed Murray a route back with three double faults in the eighth game and the Russian twice needed to hold serve at 4-5 and 5-6 to avoid losing the second set, which went to a tiebreak.
It proved to be a mini-blip from Rublev, who regained his composure to close out victory and get his hands on the crown for the first time.