Korda had trained with Nadal earlier this week before the draw paired the two together – an eagerly awaited encounter for the younger man.
Korda is such a big Nadal fan his cat is named after the Spaniard, but he caused some concern for the 21-time major champion and his other supporters on Saturday.
Having talked up the meeting, it looked as though the occasion might get the better of Korda as he quickly fell 4-0 down in the opener.
Nadal had not dropped a set since victory at the Australian Open – one of three tournaments he has entered this year and three tournaments he has won.
But that perfect 15-0 record suddenly came under threat in the second set, as Korda sought to prolong his dream match-up and stunned his hero 6-1.
Suddenly, Nadal was forced to face down one of his biggest fans in a decider and initially struggled badly, falling two breaks down, with Korda serving for the match at 5-2.
This time the nerves did get to Korda, who lost four games in a row before stopping the rot to reach a tie-break.
There, Korda did briefly hold a mini-break lead, but that was only as part of a sequence of five consecutive mini-breaks that took the match away from him, Nadal prevailing 6-2 1-6 7-6 (7-3).
"I feel very, very lucky today to be through, honestly," Nadal said afterwards.
Korda could at least enjoy his consolation prize: compliments from Nadal, who had won their only prior meeting en route to his 2020 French Open title.
"Sebastian was playing some fantastic tennis and I'm sorry for him," Nadal said. "He had chances, but that's tennis.
"He's very young, he has an amazing future. I wish him all the best."
Up next for Nadal is Dan Evans, whose compatriot Cameron Norrie – the defending Indian Wells champion – also advanced with a straight-sets win over Pedro Martinez.
Kontaveit was the second-highest seed remaining in the tournament entering Monday's play, and appeared to be on her way through to the fourth round after breaking twice in the opening set.
Yet Vondrousova got the better of her Polish opponent in a topsy turvy second set, forcing a decider that starter in frantic fashion, with a concession of serve in each of the opening five games.
Vondrousova held to end that street yet was pulled back to 4-4 by a resurgent Kontaveit, though the Czech world number 33 eventually closed the match out at the fourth time of asking in the tie-break to tee up a clash with Veronika Kudermetova.
The 21-time grand slam winner is set to sit out the Indian Wells Open and Miami Open after the US extended the requirement for all non-American nationals to be vaccinated.
Djokovic's participation at August's US Open may also be in doubt should those requirements, which are currently in place until April 10, continue to be extended.
But having also been forced to miss events in the likes of the US, Canada and Australia last year, Djokovic accepts he may have to do the same in 2023.
"I don't think there's anything official yet, so when it is we can speak about it," Djokovic told reporters on Thursday when asked about the US restrictions.
When told the rules have been certified by the US Transportation Security Administration, Djokovic replied: "I mean, if it is official then it is – what can I do? Nothing.
"You know my position, so it is what it is. I'm hoping [to play], but if I can't go, I can't go."
Djokovic, cleared to compete in Australia this year, was speaking after defeating Quentin Halys 7-6 (7-3) 7-6 (7-5) in the last 16 of the Adelaide International.
The 35-year-old, who beat Constant Lestienne in straight sets in his opening singles match of 2023, trailed 5-2 in the first set but recovered well and then edged the second set.
"It was a great performance from my opponent today and I want to congratulate him for a great fight," Djokovic said in his on-court interview. "He played like a top-10 opponent.
"He was serving big, hitting his spots and on a fast court like this it's tough to break, so two tie-breaks is a realistic score in today's match. I'm glad to overcome the challenge."
Djokovic will now face Denis Shapovalov, who saw off Roman Safiullin 6-4 6-3, for a place in the semi-finals.
"Denis is one of the most complete players out there," Djokovic said of his next opponent. "He has a very dynamic style of tennis.
"I haven't played him for a while so I'm looking forward to it. Every match from here gets tougher."
Sebastian Korda also advanced on Thursday with a 6-4 6-4 victory against Roberto Bautista-Agut, while Jannik Sinner beat Thanasi Kokkinakis 7-6 (7-2) 6-4.
At the Maharashtra Open in Pune, top seed Marin Cilic withdrew from his quarter-final with Tallon Griekspoor due to a knee injury.
World number 95 Griekspoor will now face Aslan Karatsev for a place in the final.
Raducanu saw her 2022 season cut short by a problem with her right wrist, while a bout of tonsillitis caused her to miss last week's Austin Open.
The 2021 US Open winner was seen practising with strapping on both wrists on Wednesday, ahead of Thursday's Indian Wells opener against Danka Kovinic.
Asked about her fitness, Raducanu told BBC Sport: "The same issue that I had last year has started to flare back up.
"I've definitely been managing my load. It's something that has just come back, so I'm trying to be as cautious as possible. I'm going to do everything I can to be there tomorrow."
Raducanu's recent struggles with tonsillitis represented the latest setback of a year in which she has played just four competitive matches, and the 20-year-old has been frustrated by her lack of time on court.
"It affected me badly for the short term, so I haven't prepared much. I've probably never felt so sick in my life," Raducanu reflected.
"When it rains, it pours. This [the wrist injury] and the infection… it's obviously a challenge. My preparation hasn't been ideal. I played on the weekend and then I rested it. It's how you manage things when you aren't prepared."
Alexander Zverev's struggles continued at Indian Wells as the top seed was sent crashing out of the tournament in his opener on Friday.
