The opening-round clash in Madrid marked Murray's first Davis Cup match in three years, and proved a tough test for the Scot, who lost the first set narrowly in a tiebreak -- in which he held a set point -- before surrendering a 3-1 lead in the second frame.
Murray, 32, showed his signature grit to clinch the set 6-4, however, and demonstrated even greater resolve in the decider.
The three-time Grand Slam champion trailed 4-1 in both the set and the tiebreak, but clawed his way back both times, eventually edging out Griekspoor after nearly three hours on court.
"I don't think I deserved to win that match," Murray said after his victory.
"I fought extremely hard, but he dictated a lot of the points. I thought he served amazing.
"I just fought hard and scrambled at 4-1 down in the tiebreak. It was enough to go my way.
"We knew he had been a bit injured -- he hurt his ankle four weeks ago. We had sort of prepared to play against the No. 2. An hour before the match it changed, [so] I didn't know an awful lot about his game."
In the second singles match of the first-round tie, British No. 1 Dan Evans suffered a three-set loss to Robin Haase after the Dutchman came through 3-6, 7-6 (5), 6-4 to keep his nation's hopes alive.
Victory for Britain against Kazakhstan in their second round-robin tie on Thursday would see them through to the quarterfinals as group winners.