Dan Evans lost to Robin Haase in the second singles rubber before the tie-concluding doubles rubber, in which Jamie Murray and Neal Skupski daw off Wesley Koolhof and Jean-Julien Rojer in straight sets.
"I’m obviously relieved, I don’t think I deserved to win," Andy Murray said.
"I fought extremely hard, but he was dictating a lot of the points. I thought he served amazing."
Murray, who looked laboured throughout the contest, had his trademark defence to thank after edging the tightest of finales.
At 4-3 behind in the breaker, the 32-year-old adeptly chased down two balls he had little right to keep in play and when it came back a third time, Griekspoor overcooked it and it proved to be the decisive moment.
Andy Murray of Great Britain speaks to Great Britain captain Leon Smith during his Davis Cup Group Stage match against Tallon Griekspoor of the Netherlands during Day Three of the 2019 Davis Cup at La Caja Magica
Image credit: Getty Images
However, the Netherlands were to strike back in the second rubber of the tie, with Robin Haase coming back from the brink of defeat against Dan Evans.
It was the Brit who looked like the vastly superior player for much of the rubber, racing into a 3-0 lead in the first set from which Haase never recovered.
When he broke in the seventh game of the second set, Evans had the chance to serve it out, only for Haase to break back and level the match in the eventual tiebreak.
Haase began the final set with the wind in his sails, breaking Evans in the first game and though he missed the chance to go 3-0 up, the deficit proved to be insurmountable for GB and the Dutchman held his nerve to level the tie, despite a late surge from his 29-year-old opponent.
It all came down to the doubles rubber. Jamie Murray and Skupski broke in the penultimate game of a tight first set to take it 6-4, but soon found themselves 4-1 down in the second thanks to some lax serving from the Scot.
However, with the Dutch serving for the set, Britain struck back and summoned another comeback to save two set points in the tie-break to win 8-6 and seal victory in the tie.
"It's one of the longest days I've had sitting in the chair. It's been another day of incredible drama. You can't put rankings on it as everyone plays at different levels. The Netherlands did themselves proud and played their hearts out but so did our guys," Team GB captain Leon Smith said.
"Andy [Murray] didn't feel great and came through. Dan [Evans] could have been shaking hands with a 2-0 victory but it wasn't to be. Jamie [Murray] played some unbelievable stuff and Neal [Skupski], to play his first tie, shone today.
"They [Murray and Skupski] combined brilliantly and I'm so pleased for them. Dan [Evans] came out straight away and was on his feet more than anyone else. It's not about one person, it's about the team pushing each other on. We will need that again tomorrow."