The Spanish world number one, 31, extended his Open Era record to 42 successive sets won on his favourite clay surface.
The 10-time champion will play Belgium’s David Goffin in the semis.
Greece’s Stefanos Tsitsipas, 19, beat world number seven Dominic Thiem 6-2 6-1 for the biggest win of his career.
World number 63 Tsitsipas punished an error-strewn display to set up a semi-final against Spanish world number 11 Pablo Carreno Busta, who also caused an upset by beating second seed Grigor Dimitrov 6-3 7-6 (7-4).
Goffin beat Spain’s Roberto Bautista Agut 6-7 (3-7) 6-2 in a match that lasted two-and-a-half hours.
Nadal saved three set points against Klizan, who beat Novak Djokovic in the second round.
Klizan, world’s 140th was bidding to become the first player to beat Djokovic and Nadal in the same clay-court tournament.
After winning only 13 points in a 31-minute opening set, Klizan rallied in the second and became the first player since Thiem last May to take five games off Nadal in a set.
But he could not hold on to the advantage. When attempting to serve out the set as Nadal broke back for 5-5 and won the next two games. This sealed his 17th clay-court win in a row.
This clay season has proven to be Nadal’s greatest strength. We are looking forward to the semis as well as the finals to see who will rank second after Nadal.
We continue to experience surprises on the clay court from players such as Tsitsipas. He managed to get Thiem out of the way.
We’re waiting with great anticipation to see what’s next for our players. Nadal has posed quite a challenge and is proving unstoppable. It is now obvious that he will win the Barcelona Open Final without a hitch.