tiebreaker to win as he struck an overhead winner while Murray netted a down the line backhand return before double faulting to end the set.
Murray took set three 6-3 before falling behind 0-5 in set four.
Tsitsipas took set four then left the court for a controversial bathroom break which lasted eight minutes.
Upon Tsitsipas' return to the court, Murray appeared bothered and annoyed by what he perceived as a stall tactic from his opponent.
"It's just disappointing because I feel it influenced the outcome of the match," Murray said of Tsitsipas' bathroom break.
"I'm not saying I necessarily win that match, for sure, but it had influence on what was happening after those breaks.
I rate him a lot. I think he's a brilliant player. I think he's great for the game. But I have zero time for
that stuff at all, and I lost respect for him."
In the final set, the Greek broke Murray's opening service game before taking leads of 4-2 and 5-3.
Murray held serve to reach 5-4.
Serving for the match, Tsitsipas struck an inside out down the line forehand winner to lead 15-0.
Murray netted two down the line backhands to end the match.
Afterwards the two players engaged in an icy handshake at the net.