The two-time Wimbledon champion made an impressive start but could not keep up the same pace as he was edged out 1-6 6-3 10-8 by the current British No.1.
Murray led 4-1 at one stage in the match tie-break but his composure ebbed away, the Scot particularly angry at one point by the movement of a member of his support team, who was in his eye line off the court.
"Stop moving back there," Murray shouted.
Evans stuck to his task and a wayward Murray forehand gave the Englishman two match points.
Murray saved the first with a big serve but then saw Evans get lucky on his second opportunity, a backhand that looked to be going out striking the net and bouncing in.
Murray was competing for the first time this year after pelvic problems kept him out of the opening weeks of the season, with tennis then suspended from March because of the coronavirus.
This has been a behind-closed-doors indoor tournament, and Murray has at least had good match practice, playing four opponents in a week, winning twice and only losing to Kyle Edmund and Evans on tie-breaks.
Evans said: "He did a lot good in the first set and in the end it came down to the big points.
"I was very lucky on match point but I'm just happy to come through.
"I'm surprised by how well he was executing. Not so much the aggression but I was really surprised, he hardly missed a ball for set one.
"You always know with Andy he's going to come out with some top tennis."
Evans, ranked No.28 in the world, thought his own efforts in London would be forlorn when Murray sped away in the tie-break.
"You're thinking the worst of course. I hung in there, but that's how it goes at times," he said. "I'm just really happy to come through."