World number 17 Garcia reached her only other Grand Slam last-eight at Roland Garros five years ago.
The 28-year-old is in a rich vein of form, having arrived in New York on the back of a title run in Cincinatti and is now on a 12-match win streak.
"Alison is a very tough player, a great fighter, great mover but I was ready for it," said Garcia, who has yet to drop a set at the tournament.
"The first set was not great tennis but I managed my emotions better in the rest of the match."
She added: "I am so excited, it's been a great few weeks in the US. I am going for my shots and having fun."
Garcia had lost all three of her previous meetings against Riske-Amritraj but was always in control on Sunday.
She broke in the 10th game of the first set to seal the opener and raced to a double break to lead 5-1 in the second.
The powerful Frenchwoman wrapped up victory in the next game on a fifth match point having fired a total of 30 winners compared to the American's seven.
Garcia will face either Coco Gauff of the United States or China's Zhang Shuai for a spot in the semi-finals.