The 18-year-old Hungarian was twice a break behind in the opening set but Vesnina, the BNP Paribas Open champion, could not make the most of all the break points she had as Stollar responded with 13 aces to keep herself in the match.
She edged ahead in the tie break and assumed control, leading for most of the match, but appeared unable to get over the line as Vesnina, a two-time Charleston finalist, saved five match points at 5-4 in the second set to force another tie break.
However, Stollar prevailed once more and took victory in two hours and five minutes to set up a clash with fellow youngster Jelena Ostapenko, after the Latvian took out top ranked Greek star Maria Sakkari 6-2 4-6 6-2.
Daria Kasatkina maintained her first-round form to overcome Puerto Rico's Monica Puig 6-0 6-7 (3/7) 6-2.
Kasatkina bagelled the Olympic gold medallist and went a break up in the second set before Puig got on the scoreboard in the ninth game of the match and went on to lead 5-3. However, the Russian broke serve and forced a tie break but Puig drew level.
Puig had the first break of the decider but ultimately unforced errors cost her and it was one such mistake on match point that saw Kasatkina go through. The former French Open junior champion will face good friend and doubles partner Daria Gavrilova after the number nine seed beat American Alison Riske 6-3 6-1.
There were also straight sets victories for Naomi Osaka, Kiki Bertens, Mirjana Lucic-Baroni, Ons Jabeur and Anastasija Sevastova.