Britain were looking to follow up their opening Group D win over Australia at the inaugural mixed international team competition and had been given the perfect start following Norrie's 3-6 6-3 6-4 victory over Nadal for the biggest win of his career.
Swan, ranked 145 in the world, then produced another fine display as she recovered in her match against Parrizas Diaz after the Spaniard had taken the first set.
Encouraged by team captain Tim Henman, the 23-year-old rallied to defeat the world number 70, coming through 3-6 6-1 6-2 for her first win in the competition.
Spain will now need to mount a recovery of their own in Sunday's fixtures to stay alive in the tie, while victory will see Britain qualify for the quarter-finals.
Paula Badosa, the world number 13, will take on Harriet Dart, who beat her at the Billie Jean King Cup, while Pablo Carreno Busta faces British number two Dan Evans.
Should the tie end up level at 2-2 after the singles, it will be decided on mixed doubles, which would see Nadal and Badosa play Evans and Dart.
Norrie had lost his four previous meetings against Nadal, but produced some of his best tennis at a sold-out Ken Rosewall Arena on Saturday to give Britain an early lead.
Asked if it was his best career win at a press conference, Norrie said: "I think it was, especially on ranking and beating a guy like Rafa as the competitor that he is. I know it's his first match of the year, but it was a sick win.
"He absolutely chopped me the last five or four times I played him, so it is nice to get him once now. I really played well and held my nerve throughout the match."
Nadal was making his first appearance since the Nitto ATP Finals in November.
The 36-year-old world number two felt that despite the defeat, it was "not a disaster at all".
Nadal said: "I can do things better of course, and I need to do it, but as I said, I had not many matches before. It's all the credit to him that he did a lot of things very well.
"I need to be a little bit faster physically, a little bit more solid, some less mistakes and making better decisions at some point, playing a little bit longer.
"I think there is a way to improve, but I have time before the Australian Open starts in two weeks."
Nadal, though, offered no suggestions he was ready to follow great rival Roger Federer into retirement just yet.
"When the day arrives that I go out on court and say 'that's getting hard', is the day to say goodbye. I don't want to and I don't need to keep playing for no one reason," the 22-time Grand Slam winner said.
"The only reason that I make sense to keep playing is if I really enjoy what I do and if I really feel it. If I don't feel it, I mean, I did much more than what I ever dreamed."
Nadal added: "But either way, for me, I lost my match. That's it, no? Every time that I am coming to a press conference seems that I have to retire.
"So you are very, very interested on my retirement? That is for the moment, not the case.
"When this day arrives, I'm going to let you know. Don't keep going with the retirement (questions) because I am here to keep playing tennis."
Elsewhere in Saturday's matches, Croatia are 2-0 up on Argentina in Group F in Perth, while it is 1-1 between Belgium and Bulgaria.
In Sydney, Germany lead the Czech Republic 2-0 in Group C and Brazil hold a 2-0 advantage over Norway from their Group E match in Brisbane.