The two-time Wimbledon champion saved eight break points in the third to keep the contest alive on the same court where he collected his maiden major title a decade ago.
While he could not hold off his opponent's offensive onslaught in the fourth, the performance nonetheless gave him hope after hip surgeries in 2018 and 2019.
"I've got a metal hip. It's not easy playing with that. It's really difficult. I'm surprised I'm still able to compete with guys that are right up at the top of the game," he told reporters.
"Matches like this, you know, I'm really proud that I have worked myself into a position where I'm able to do that ... it's been a difficult six years for me. It's been really hard."
Murray produced impressive wins over Greek Stefanos Tsitsipas and Australian Nick Kyrgios in June's Stuttgart tournament but suffered an abdominal injury when he lost in the final to Berrettini.
The performance put him back in the top 50 for the first time in roughly four years - though he was ranked 51st in the world headed into Flushing Meadows - an improvement he said helped motivate him.
"Lots of people told me I wouldn't be able to play again, and lots of people told me I'd be able to hit tennis balls but not compete professionally again," said Murray.
"That was nonsense, and I want to see how close I can get back to the top of the game."
