After saving eight match points in four matches, including five in the semi-final to beat Jiri Lehecka, Murray knew he had his work cut out against No 3 seed Medvedev.
And so it proved, for despite breaking back in both sets Murray fell 6-4 6-4 to his fellow former world No 1.
“It was an incredible week. Obviously it didn’t finish how I wanted but I had some amazing matches here and created some great memories. It was fantastic to be in a final again,” said Murray, who is set to rise 18 places in the rankings to No 52 on Monday.
“Daniil is one of my favourite players to watch and one of the best players on the tour. It was great for me to get that opportunity to play against someone of his level in a final again. Definitely some things to work on.
“I am proud of my week and the work me and my team have put in to get me back to this stage.”
Murray’s last-gasp exploits had not gone unnoticed this week, with the 35-year-old laughing after the final when Medvedev said: “I was like ‘Oh my god I have a match point against Andy — that’s not a good sign.’”
Next up for Murray is the ATP 500 event in UAE, the Dubai Duty Free Tennis Championships which starts on Monday, meaning the three-time grand slam champion is just days away from his next match.
Murray, who won this event in 2017, faces a difficult opener against No 5 seed Hubert Hurkacz, with Novak Djokovic a potential quarter-final opponent should Murray win his first two matches.
That is not out of the question for Murray, who beat Hurkacz in their last meeting two years ago in Vienna, with world No 84 Alexei Popyrin the possible second-round opponent.
Medvedev, meanwhile, is hoping to build on his 17th Tour title.
“[It was] a tough match,” said Medvedev after his win. “It was a little bit windy today, so both of us were struggling to find our rhythm… Today was a big fight for both of us. Sometimes we were playing bad, then suddenly both of us were playing amazing, and I’m happy to win.
“I managed to find momentum in Rotterdam. The first two matches I played there were tough. I didn’t feel good before that tournament, and then starting in the quarter-finals against Felix [Auger-Aliassime], I started to find my rhythm.
“I started to play really good. Serve better, move better. Today it was tough for Andy to put the balls through me many times, and now the question is [if it] can continue. I’m going to try as long as I can.”