Murray came from a set down and then trailed 4-1 in the decider in Madrid before beating the spirited Griekspoor, ranked 179 in the world, 6-7 (7/9), 6-4, 7-6 (7/5).
A scrappy victory gave Great Britain the lead in their opening Group E tie ahead of the second singles rubber between Dan Evans and Robin Haase, and then a third match in the doubles.
Murray had given short shrift on Tuesday to those he believes have been too quick to criticise the revamped Davis Cup, which for the first time is featuring all 18 World Group countries competing for the trophy across a single week in one venue.
And in what will be music to the ears of the organisers, among them Barcelona footballer Gerard Pique, Murray was full of praise for the atmosphere inside the Caja Magica's cosy Stadium 3.
"The atmosphere was brilliant," Murray said afterwards. "That was the one concern I had about the event, about it being on neutral ground, but it was great."
British fans have been second only to Spanish in terms of tickets sold so far and the strong attendance for Murray's opener inside the tournament's smallest court, with a capacity of 2,500, has not been consistent across other morning matches.
Murray, along with Rafael Nadal and Novak Djokovic, is one of the competition's biggest attractions this week and Britain's prospects of progressing to the quater-finals look good from a group that also includes Kazakhstan.
It remains to be seen how Murray's contribution is managed, both in singles and doubles. The three-time grand slam champion had not played since winning the European Open in Antwerp on October 20, a significant achievement for the 32-year-old who is still regaining fitness after undergoing hip surgery in January.
He was given a stiff workout by the 23-year-old Griekspoor, who has never played in the main draw of a grand slam and who was a surprise pick in the Dutch team ahead of Botic Van de Zandschlup.
"About an hour before the match it changed," said Murray. "I didn't know much about his game."
Griekspoor saved a set point with an ace in the first-set tie-break before converting his own second set point with a forehand winner.
Murray levelled up with two breaks in the second set but his opponent stormed 4-1 ahead in the decider and then, after Murray fought back, 4-1 clear in the tie-break.
But Murray again dug deep, his experience and resilience proving the difference as Griekspoor lost his nerve and then the match.
Djokovic enjoyed a more comfortable start as he cruised to a 6-1, 6-2 victory over Yoshihito Nishioka to give Serbia an unassailable 2-0 lead over Japan, who came close to upsetting France on Tuesday.
Filip Krajinovic had earlier defeated Yuichi Sugita in straight sets.
Argentina could have secured progress by beating Germany but they lost both of their singles rubbers as Philipp Kohlschreiber beat Guido Pella and Jan-Lennard Struff prevailed against Diego Schwartzman.