The Ukrainian appeared to be heading for defeat at Indian Wells after losing the opening set and requiring an eight-minute medical timeout.
However, she recovered impressively to claim a 3-6, 6-2, 6-2 victory and remain in contention for a place in the semi-finals.
Svitolina, who suffered a heavy 6-4, 6-0 defeat to Mirra Andreeva in her opening round-robin match, told Sky Sports: “It was not an easy task to do.
“I was pretty angry at myself, after the first match and after the first set also. Things didn’t go my way, so I tried to direct this anger towards the ball and found my game in the end.
“I think the level was back in the third set. I just tried to give it all in this match because I know I would be eliminated if I lost. So I just tried to really release all this energy and find these openings against Karolina. She’s such a tough opponent.
“It’s the last tournament of the season, so I just have to really put my head down and deal with all the small injuries.
“I mean, it’s normal that everyone is tired. Everyone is struggling with something here and there. I will just try to do my best, have an ice bath and try to heal as much as possible in the next 48 hours.”
Svitolina will now face Aryna Sabalenka in a winner-takes-all clash for the group’s remaining semi-final spot.
Mirra Andreeva has already secured her place in the last four after recovering from a 4-1 deficit in the opening set to beat Sabalenka 6-4, 6-0.
Andreeva leads Group B, with Sabalenka in second place and Svitolina third heading into the decisive encounter.
