The world number 114 levelled the World Group first-round tie at 1-1 following a stirring display for Great Britain in Marbella on Friday.
He trailed by two sets and was 3-1 down to the world number 23 before roaring back to win 4-6 3-6 6-3 6-2 6-2 in his first match on clay.
"It's 100 per cent the best day of my career," Norrie told the BBC on court after the game.
"I was nervous and had a streak in the second set where I couldn't get a serve but I pulled it together.
"Thanks to my team, they gave me great energy. I thought I was tougher and physically I had some problems but I played it point by point.
"I managed to pull it together and keep Great Britain in the tie. It gave me a lot of confidence because it was my first match on clay."
Norrie broke three times in the third set and twice in the fourth and fifth and Bautista Agut had no response with the Briton hitting 60 winners - twice as many as his rival.
Norrie outfought Bautista Agut, who began to suffer with a thigh problem in the fourth set.
The 22-year-old added: "It was a little boost but I backed myself physically and mentally. It was pretty good of him not retiring but I wanted to keep the points physical and work him as hard as I could.
"I talked to Kyle Edmund a lot on how to manage the energy, you have to pick your points and he is very good at that. I'm a good runner and had to back myself the whole way. The fans helped and in the fifth set I went to serve and there was hell of a noise."
GB captain Leon Smith also told the BBC: "It's amazing to see Cam, as a rookie, a debutant, actually wear the opponent down. We've had a lot of good moments in the Davis Cup and this is one of them right now.
"I'm almost a bit lost for words. All you ask for is that they give their best effort and fight like hell. Cam fully deserves every bit of praise he gets."
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