The Briton, back on the ATP Tour after a career-saving hip resurfacing operation in January, was on course for a straight-sets victory but Copil was determined to test his opponent's resolve and clawed his way back to force a third set.
Murray came through with flying colours, converting five out of 14 break points and winning 87% of points on his first serve in the match that lasted two hours and 35 minutes to reach his first singles semifinals since the French Open in 2017.
After cruising through the first set, three-times Grand Slam champion Murray raced ahead 3-1 in the second before suffering a lapse in concentration while serving for the match at 5-3.
A double fault by the Briton handed Copil the break and the Romanian took full advantage to drag the contest into a decider after edging the second set in a tiebreak.
The pair were locked at 4-4 in the final set before Murray switched gears to close out the victory in style, finishing it off with his ninth ace.
"I haven't played loads of matches the past few years, so when you get to the end of the match, it's always difficult to serve it out," Murray said in a courtside interview.
"I played a bad game at 5-3 in the second set and after that I think he gained a lot of confidence. He served extremely well.
"He was being a lot more aggressive at the end of the second set and in the third, but thankfully I managed to get the break right at the end. It was a tough one to get through."