It was the British number three who had his serve broken first in the third but he responded when Basilashvili served for the set.
The Georgian did not recover and from there lost nine of the last 10 games as Edmund set up a clash with 15th seed John Isner or Australia's John Millman.
Meanwhile, Naomi Broady's French Open debut ended in a battling defeat to American Coco Vandeweghe.
Broady became the third British woman inside the top 100 in February and earned her place at a grand slam on ranking for the first time at Roland Garros.
Clay is her least favourite surface but the same can be said for fellow big server Vandeweghe and it was a close contest that the British player eventually lost 6-4 3-6 6-3.
Broady, from Stockport, began confidently and led 4-2 but from there she lost five straight games.
She recovered to win the second set and was 2-0 up in the third but another run of five successive games for Vandeweghe proved decisive.
It has nevertheless been a positive experience for Broady, and the 30,000 euro she will collect will mean she is at last making money from tennis.
Roland Garros French Open Tennis 2016 - In pictures
