For the third successive match, Robredo fought back from two sets down, matching a feat achieved only once before in grand slam tennis, by Frenchman Henri Cochet at Wimbledon back in 1927.
Robredo, though, said: "I don't really care, frankly. What is very important is that today I won this match.
"I won a big match against a player who defeated me five times. And the rest is only records.
"I played a player who is incredible on clay. I'm not thinking about history. History is this match I played today. It's not the score that counts."
Almagro, the 11th seed, looked in complete control at two sets up and was a break ahead in the third, fourth and fifth as well, but he could not put his fellow Spaniard away and the 31-year-old fought back to win 6-7 (5/7) 3-6 6-4 6-4 6-4.
Robredo sobbed into the clay as the crowd chanted his name, and he admitted, at 4-1 down in the third set, he could not have imagined what was to come.
He said: "Maybe he had a little bit of doubt in that moment. And then I just pushed hard and I won the third. And then I was just dreaming and dreaming to try to do it again, and I did it.
"I know players like me, when you dream, you always dream to win the trophy or to win the match, but you never realise that you are going to win being two sets down and 4-1 down, and 4-2 in the fourth and 2-0 in the fifth.
"That's too much. That's too many hours thinking. And, me, I prefer to think I win 6-3 6-4 6-3 and then the dream, it finishes quicker.
"But, anyway, it's a lovely dream. And hopefully I can remember it a lot of times because during the year you don't have that many experiences like that."
Robredo's achievements are all the more noteworthy because this time last year he was returning to the sport after seven months out following hamstring surgery and ranked 470th in the world.
He said: "Since I got the surgery, I've never had this pain again. I just repaired my body and now I'm okay.
"I already forgot that. I already erased that from my mind. Obviously it was an experience that I had to go through, and in life all the experiences are good. I think that you learn a lot more from a bad experience than from a good one."
Robredo next plays fellow Spaniard and fourth seed David Ferrer, who continued his cruise through the draw with a 6-3 6-1 6-1 victory over South Africa's Kevin Anderson.
Ferrer is into his sixth consecutive grand slam quarter-final and hoping to reach his first final, but he said: "Tennis doesn't owe me anything. Tennis is one of the fairest sports. It's given me so many extraordinary feelings."
Also through to the last eight is French favourite Jo-Wilfried Tsonga, who has looked in fine form so far in Paris and beat Serbia's Viktor Troicki 6-3 6-3 6-3.