The 6-3, 6-7 (6), 6-1 victory was Murray’s first on the ATP Tour since he beat James Duckworth in Brisbane on 1 January while battling right hip pain that prompted him to undergo a potentially career-ending operation less than a month later.
Murray faced American Sandgren in the second singles match of his comeback last month, losing in two close sets, and this was a demonstration of the progress he has made since then as he moved well, served strongly and dictated much of the play.
The former world No 1 said: “I think that, in some ways, it’s one of the best wins that I’ve had, not in terms of just getting through that today, but everything that’s gone into getting back to this point.
“You don’t necessarily appreciate what it’s like to be healthy and stuff at the beginning of your career, it’s something that you take for granted and it’s quite easy.
“Whereas these last few years haven’t been and obviously after the operation in January it’s been difficult and it was undecided at times about whether I wanted to keep going or not and it’s been tough. But I’m really glad that I can actually get the win today and hopefully I’ll be able to keep going.”
Murray played his first tournament at second-tier Challenger level in 14 years in Majorca while his rivals were doing battle at the US Open and has been training hard since ahead of four consecutive tournament weeks. His aim is to win two or three matches at each of them and, on this showing, that is certainly realistic.
Murray, who did not face a break point during the match, took the first set with a break of serve in the sixth game and had a match point at 6-5 in the second-set tie-break.
But Sandgren, ranked 69, saved it and made it three points in a row to take the contest to a third set.
In humid conditions, this was a test of Murray’s physical condition but it was Sandgren who was found wanting.
After Murray broke to lead 3-0, the American took a long medical time-out to have strapping applied to his foot and then dropped serve again after a mammoth game.
Murray was unable to take two chances to finish the match with a bagel set but polished things off in his next service game and will take on Australia’s Alex De Minaur in the second round tomorrow.
The Scot said: “Tonight I was hitting the ball well, movement was I think pretty good. I served well.
“Obviously there’s things that I feel I can do better, but it was definitely progress.
“I don’t know whether that’s five per cent, 10 per cent, it’s difficult to put a number on it, but I think I did a little bit better tonight than I did when we played a few weeks ago and that’s a good thing.
“That’s what I need to keep doing the next few months and then I’ll see – at some stage the progress will stop and I’ll find out what my limit is. But it’s good to see that I’m still progressing and making improvements.”