The world number one was 4-0 up in the deciding set against Albert Ramos-Vinolas before the Spaniard hit back to qualify for the quarter-finals with a 2-6 6-2 7-5 victory.
Murray was returning from injury at the tournament having been sidelined with an elbow problem since March 12.
And the 29-year-old told Sky Sports: "If I'd been offered this result a week, 10 days ago, I would've taken it.
"I didn't know if I was going to be able to play here or not.
"For me personally - it's not going to seem it to everyone watching - but it's progress from where I was 10 days, a couple of weeks ago.
"I'm going to need to put some work in, there are something things I need to improve upon. But it wasn't too bad."
Murray's serving woes continued as he was broken seven times during the see-saw encounter in Monaco.
The Scot started his second-round match against Gilles Muller on Wednesday with three double faults in the first four points of the match and it did not get much better in his match-up against clay-specialist Ramos-Vinolas.
Murray, however, was not too concerned with his serve.
"If you're serving for two weeks 30, 40, 50 kilometres an hour slower than what you're used to, you're going to serve slightly differently and then when you start serving fast again it just takes a bit of time to get into a rhythm," he added.
"I felt like - even though I didn't hold serve that much today - that I served better than I did in my last match. Second serve had a little bit more speed on it, less double faults, a few more aces.
"That's something that I think will keep getting better.
"I was really happy with how my elbow felt this week. It got better each day. I'm going to need to obviously improve my serving over the next few weeks."