Reigning Wimbledon champion Murray raced through the opening set against the unseeded Kazakh, dropping just the solitary game in 34 minutes.
The seventh seed continued his high standard of tennis throughout the second stanza, mising his trademark defensive prowess with some wonderful shot-making to take a 2-0 lead.
However, Golubev fought back to win the third set, with Murray looking out-of-sorts, though that was all the Kazakh was afforded, with the Scot advancing in 2 hours and 35 minutes.
"There have been quite a few upsets here the last few days and tricky conditions," Murray, who missed the tournament last year with a back problem that eventually required surgery, told reporters.
"So the most important thing was to get through."
"I think I've done a good job of that the last few years," added the Scot who has not lost in the opening round of a grand slam tournament since a defeat by Jo-Wilfried Tsonga six years ago at the Australian Open.
"I have not always played my best tennis at the beginning of the tournaments, but I found ways to get through and get myself into the tournament and give myself opportunities to do well in these events."
Without a career title on clay Murray's chances at the French Open are probably less than at the other three slams, but he played impressive tennis against Golubev at times and defended like a contortionist when pushed into corners.
With a semi-final and two quarter-final runs at Roland Garros to his name it will take a good player to beat Murray here, although he knows he will not be able to afford the kind of lapse he suffered in the third set against Golubev.
Despite dropping his serve too often for his liking and being outplayed for a while in the third, Murray rarely looked panicked and finished with something to spare.
Next up is Australian Matosevic who finally won a match at a grand slam at the 13th time of asking on Tuesday, beating dreadlocked German Dustin Brown.
"He's an interesting character, that's for sure," Murray said. "He's a good guy and he's good fun, good fun around the locker room, always makes everyone smile, makes everyone laugh."
Funny or not Matosevic will probably find the Mr Reliable of the grand slams a far too serious proposition on Thursday although he will feel a good deal lighter on the court.
"I feel like there is a huge gorilla off my shoulders," he said. "It's just relief more than anything else really.
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