Last month Murray admitted he was in a “terrible moment” but said he “won’t quit” in response to an article which questioned when the Scot should retire.
Against Czech Machac, 23, he was broken in his second serve game and, having recovered to 5-5, gifted his opponent another break for 6-5 with a double fault and backhand error.
Murray lost his serve again early in the second and missed an opportunity to break back at 3-2 before Machac comfortably closed out one of the biggest wins of his career.
Elsewhere in Abu Dhabi, Naomi Osaka’s comeback was halted by another first-round defeat as she lost 7-5 6-0 to American Danielle Collins.
The Japanese former world number one returned last month after giving birth in 2023, but struggled after losing a tight first set.
Osaka won her first match back at the Brisbane International at the start of January but has lost her three matches since, including against Caroline Garcia in the Australian Open first round.
“I know she’s still only a couple of matches into her comeback,” said Collins, who is currently ranked 71st in the world.
“But I think she’s playing well.
“She played a good first set and was serving very well and doing some really great things out on court.
“I’m sure at this point it’s just kind of regaining confidence and getting more matches under her belt and getting back into the routine of tournaments and that stuff, which can be challenging. But I thought she played well overall.”
Britain’s Harriet Dart progressed to the second round of the Transylvanian Open with a 6-4 7-6 (7-0) win over Hungarian Anna Bondar.