World number 131 Podoroska had never won a Grand Slam main draw match before the tournament and is the only the third female qualifier to make the last four of a major.
Podoroska, 23, will play Polish teenager Iga Swiatek who defeated Italian qualifier Martina Trevisan 6-3, 6-1.
Podoroska will now attempt to become the first women's qualifier in history to advance to the final of any of the four majors.
Alexandra Stevenson reached the semi-finals at Wimbledon in 1999 while Christine Dorey achieved a similar run at the 1978 Australian Open.
"It's a little bit difficult for me to speak now after the match," said Podoroska, whose only other Grand Slam appearance came at the 2016 US Open.
"We did a very good job with my coaches during the (coronavirus) quarantine. I've been training a lot with all of my team I think that's why I'm here today."
It was the first time Podoroska had even faced a player inside the top 20, having started the year ranked a lowly 255.
She is guaranteed 425,250 euros ($501,740) following her breakout performance in Paris, easily doubling her total career earnings of $301,547.
Podoroska is also the first Argentine woman to reach the last four of a Slam since Paola Suarez in Paris in 2004.
Gabriela Sabatini, the 1990 US Open winner, is the country's only women's Grand Slam singles champion.
For Svitolina, it marked a third quarter-final defeat in as many attempts at Roland Garros, having also lost at this stage in 2015 and 2017.
The Ukrainian hit just eight winners -- the same number of times she dropped serve -- as she slumped to her worst defeat at a major in terms of her opponent's ranking.
"I had the opportunity but it didn't happen, and right now it's for sure upsetting, but in the end life continues," said Svitolina.
"I think there was many things that let me down today."
Swiatek into maiden Slam semi
World number 54 Swiatek, 19, followed up her remarkable victory over top seed and 2018 champion Simona Halep with a straight set win over Trevisan, ranked 159.
Trevisan notched her biggest win to date in the last round here by defeating fifth seed Kiki Bertens.
But she was unable to repeat the magic against Swiatek despite taking the first break of the match.
Swiatek broke the Italian six times in her confident 78-minute win.
"I did a really good job," a delighted Swiatek said.
"I used the wind really well. There were lots of errors due to the wind but I played smart."
Danielle Collins will face Australian Open champion Sofia Kenin in the quarter-finals after a 6-4, 4-6, 6-4 win over Tunisian 30th seed Ons Jabeur, in a last-16 match rescheduled from Monday due to rain.
World number 57 Collins, a semi-finalist in Melbourne last year, is through to the last eight at a major for the second time.
Collins became the fifth unseeded player in the quarter-finals of the women's draw, the most since Grand Slam tournaments were expanded to 32 seeds in 2001.
"I kind of felt like I was in the driver's seat up until 6-4, 3-0, but she's tricky," said Collins, who dropped five straight games as Jabeur fought back in the second set.
"I lost my way there a little bit. I just needed to try and stay positive."
Collins, 26, was a two-time national singles champion at the University of Virginia and turned professional after graduating in 2016 with a degree in media studies.
"I did kind of go an untraditional route and it took me a while to kind of figure everything out," she said. "I'm still trying to figure everything out on this tour."
She lost in the first round of last month's US Open but has since teamed up with former top 10 player and clay specialist Nicolas Almagro, a three-time quarter-finalist in Paris.
"It's a really special treat for me to be able to work with him," said Collins, who beat Kenin in straight sets at Adelaide in January.