The victory set up a clash with former US Open champion Del Potro, who is still working his way back up the rankings after wrist injuries that nearly ended his career.
As del Potro stepped over the net and walked toward him, Almagro collapsed on the clay court, his sobs clearly audible over the crowd's cheers of encouragement and appreciation.
Sport is not only about competitiveness but also sportsmanship and Argentine Juan Martin del Potro has every bit of it in him.
At 1-all in the third set - del Potro took the first, then lost the second - there was the startling way Almagro paused at the baseline, placed his hands on his knees and shook his head, then fell backward onto the court.
But world No.1 Murray says he's been around the block a few times and isn't going to get upset with whatever 27-year-old Klizen - ranked 50 in the world - gets up to on court.
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"It could be another great battle if I feel good", del Potro said. So I think my job is really to try to be more positive while I'm out there.
He reportedly told the world number 45 to think of his family and baby as he knelt by his side.
"And sometimes the heart is first, (before) the tennis match or the tennis life".
Del Potro will next face tennis legend Andy Murray. I wish a good recovery to Nico.
'We played great battles past year, one each, ' said Del Potro.
The 18th-seeded Aussie led by a set and 4-2 but failed to protect his advantage, falling 5-7 6-4 6-1 6-2.
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However, the Scot insisted he is playing well enough to beat Del Potro after battling past Martin Klizan. The 20-year-old reached the third round at Roland Garros for the second consecutive year.
"I'm very satisfied with the way I'm playing".
Japan's Kei Nishikori, the eighth seed, progressed by beating Jeremy Chardy 6-3 6-0 7-6 (7-5).
Czech second seed Karolina Pliskova and third-seeded Simona Halep of Romania, both firm women's title favourites after the first round exit of world number one Angelique Kerber, look for places in the last 32.
Pliskova has never got beyond the second round in six previous attempts.
Seventh-seeded Croatian Marin Cilic benefited from 11 aces to breeze to a 6-3, 6-2, 6-2 win over Konstantin Kravchuk, spoiling the Russian's debut at Roland Garros.
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Poland's Agnieszka Radwanska, seeded ninth, saw off Belgian qualifier Alison Van Uytvanck 6-7 (3/7), 6-2, 6-3.
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