Veteran Murray, bidding for his 47th career title but first since 2019, pushed the Russian hard but succumbed 6-4 6-4 in a battle of former world number ones.
Medvedev also triumphed in Rotterdam last week and now has 18 Tour-level titles in his locker.
He started strongly and was immediately into his baseline rhythm to win the majority of the long exchanges, moving into a 4-1 lead with two breaks of serve.
Murray, who saved five match points on Friday against Jiri Lehecka (21) to reach his 71st ATP final, also fell behind in the second set but hit back with a break of serve.
The British three-time Grand Slam champion had the momentum at 4-4 40-0 but Medvedev suddenly rediscovered his brick wall-like resilience to reel off five successive points, sealing a crucial service break as Murray over-cooked a volley.
Former US Open champion Medvedev double-faulted on his first match point in the following game but a superb lob completed the job when he had another opportunity soon afterwards against a weary-looking Murray.
"It was a very tough match because it was windy and we struggled for rhythm," world number eight Medvedev said on court. "Today was a big fight. Sometimes we were both playing bad, sometimes we both played well."
For Murray, it was his fourth ATP final since returning to the tour after having hip surgery in 2018 and, while he has lost three of those, he remains a force to be reckoned with.
