However, in just his sixth match on a clay court, the 114th-ranked Norrie came from behind and rallied for a 4-6, 3-6, 6-3, 6-2, 6-2 victory over the 23rd-ranked Bautista-Agut, who was struggling with a muscle strain in his left leg on Friday.
Had Norrie been able to win it was expected that Kyle Edmund, fresh from his Australian Open semi-final run, would play the deciding match for GB.
"I can't speak highly enough of what Cam has done all weekend", he said. He pushed him really close and I'm really proud of him.
The 22-year-old left-hander, who was born in South Africa and spent much of his childhood in New Zealand, showed remarkable grit against world number 21 Albert Ramos-Vinolas before bowing to a four-set defeat.
It seemed Norrie would follow him to a loss when two sets down to Roberto Bautista Agut, only to pull a rabbit out of the hat to level the tie with a 4-6 3-6 6-3 6-2 6-2 win.
It was Spain's first tie on home soil since 2013.
Injured British No 1 Andy Murray tweeted his congratulations to Norrie, now ranked 114 in the world.