Spaniard Nadal, bidding for an 11th title on the Parisian clay, also trailed by a break in the second set as Schwartzman threatened a huge shock, but after a rain stoppage, Nadal returned to court to move 5-3 ahead in the second set.
On Suzanne-Lenglen, Marin Cilic's last-eight tie with Juan-Martin Del Potro failed to burst into life until the ninth game, when the Croatian saved three break points before eventually holding, and did likewise in his next service game before taking the set to a tiebreak - which was locked at 5-5.
Having gone two sets up on the back of big hitting from the baseline, sixth seed Anderson threw Schwartzman a lifeline in the contest while serving for the match at 5-4.
The gritty Argentine went toe-to-toe with the world No. 1, breaking him three times and booking two set points.
Schwartzman, who had never even reached a Grand Slam quarter-final until last year's US Open, broke first in the second set as Nadal uncharacteristically blew a 40-15 lead, capped by two wild forehands.
They will return on Thursday morning.
Another rain delay sent the players off court and shortly afterwards play was suspended for the day.
Kasatkina will face US Open victor Sloane Stephens for a place in the semifinals.
This US postal stamp helped fund key breast cancer study
After years of follow-up, the data showed that most patients who did not get chemotherapy fared as well as those who did. Very high scores are at increased risk of the cancer spreading, and those patients have benefited a lot by chemotherapy.