And the British no 3 was staring at a straight sets defeat against Bautista Agut, but with the Spaniard struggling through injury, Norrie produced the performance of his life to secure a 4-6 3-6 6-3 6-2 6-2 victory that drew Leon Smith's team level.
Jamie Murray and partner Dominic Inglot will take to the court for GB in Saturday's crucial doubles.
A proud Murray tweeted in the immediate aftermath: "That's one of the most unbelievable wins/results/upsets I've seen in a long time on a tennis court".
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Speaking after the victory, the Brit told the media: "I was nervous and had a streak in the second set where I couldn't get a serve but I pulled it together".
He said afterwards: "In the fifth set I went to serve and there was an eruption of noise and I thought, this is an incredible experience. Well done to cam norrie and all the team".
The 22-year-old added: "It was a little boost but I backed myself physically and mentally".
It all seemed to be going world number 23 Bautista Agut's way in the second match of the day as he battled to an early two set lead. It was pretty good of him not retiring but I wanted to keep the points physical and work him as hard as I could.
Norrie outfought Bautista Agut, who began to suffer with a thigh problem in the fourth set.
Speaking on the court, he said: "At two sets down you're thinking just keep fighting, keep pride and passion". "All you ask for is that they give their best effort and fight like hell". It's awesome to see Cam, as a rookie, a debutant, actually wear the opponent down.
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