The pair, dubbed "Nadalcaraz", edged out Dutch opponents Tallon Griekspoor and Wesley Koolhof in their second-round clash in cloying heat at Roland Garros winning 6-4, 6-7 (2/7), 10-2.
The packed crowd on Court Suzanne Lenglen made no secret of their support for the Spaniards regularly chanting: "Let's go Rafa, let's go".
"We had fun because the score was positive in the (match) tie-break," said Nadal, who is chasing a third Olympic gold.
"We have been suffering but we are enjoying playing together, we are creating good synergies, good energy, so yeah, we are having fun in general terms.
"We have the positive relationship outside of the court that helps inside (the court)."
Nadal, 38, said the switch to doubles tennis was tough for singles specialists.
"We are not used to playing doubles so the only way to make that work well is to play with happiness, with energy, with high energy," he said. "That's what we are trying to do."
The red-shirted Nadal was dumped out of the singles by Novak Djokovic on Monday but emerged alongside his 21-year-old teammate to loud applause from the expectant crowd 24 hours later.
The Spaniards regularly put their opponents under pressure on serve in the opening set and finally broke in the seventh game when Alcaraz ripped a cross-court forehand winner.
French Open and Wimbledon champion Alcaraz served out as the Spaniards wrapped up the set 6-4 in 54 minutes.
The left-handed Nadal was wearing strapping on his right thigh but appeared unhindered, producing some athletic tennis at the net alongside his trademark punishing forehands.
Both teams missed chances to break towards the end of a tight second set and the Dutch dominated the resulting tie-break to level the contest.
But the pendulum swung dramatically in the match tie-break as Nadal and Alcaraz surged into a 5-0 lead, running out 10-2 winners after two hours and 22 minutes.
Alcaraz said it was a "dream come true" to play alongside his compatriot, 17 years his senior.
"Playing at the same side of the net with Rafa, learning from him, it's unbelievable.
"Learning how to deal with some situations. How he says 'Come on' or something like that -- it's amazing so I'm trying to enjoy every single second out there."
The Spanish pair will face US fourth seeds Austin Krajicek and Rajeev Ram in the last eight.
The injury-hit Nadal, a 14-time French Open singles champion, is playing just his seventh tournament of 2024 in a season in which his ranking has plummeted to 161.
He is aiming to capture a third Olympic title after his 2008 singles win and his doubles triumph in 2016.
