A day after 14-time French Open champion Nadal was heavily beaten by Novak Djokovic in the singles, the veteran bounced back with a commanding display alongside his compatriot.
The Spanish duo, dubbed 'Nadalcraz', delighted a packed and vociferous crowd on a sweltering Court Suzanne Lenglen arena as they combined their formidable firepower to keep their gold medal dream on course.
Alcaraz pummelled a forehand winner to get the crucial break of serve in the first set with Nadal roaring his approval.
Koolhof, a former doubles world number one, and Griekspoor provided stiff resistance in the second set before dominating a second-set tiebreak, winning it 7-2.
"Even if we felt today that we played much better than the first day, and in the second set we were a bit over them in terms of opportunities and level, we were not able to convert the opportunities because they played well," Nadal said.
Nadalcaraz caught fire in the deciding third set, fist-pumping and roaring after a succession of electrifying winners as they blazed to victory.
Nadal - who won the men’s doubles gold with Marc Lopez in Rio 2016 - and Alcaraz showed enthusiasm and motivation both on and off the court.
"We are not used to playing doubles. To make that work well is to play with happiness and with high energy. And that is what we are trying to do.
"We have a good relationship outside of the court. So that helps the feeling on court," Nadal said.
The pair will play their quarter-final on Wednesday evening against American fourth-seed pair Rajeev Ram and Austin Krajicek.
Earlier in the day Alcaraz will play a singles match against Russian Roman Safiullin, the world number 66, who is competing as a neutral athlete.
