The match appeared to be heading in one direction as the world number 18 broke the Brit four times in the opening set to win 6-2.
However, Dart rallied and produced three breaks of her own before claiming the second set with an ace.
The world number 122 took a sensational 3-0 lead in the decider, but Svitolina responded as she looked for a response of her own.
Despite her efforts, it was Dart who powered to victory, winning the match 2-6 6-3 6-3 in a contest which lasted for a gruelling two hours and 17 minutes.
Kaia Kanepi will face Dart next after the Estonian needed just 81 minutes to upset 22nd seed Belinda Bencic 6-4 6-3.
Liam Broady, meanwhile, failed to progress past the first round despite taking a set off Miomir Kecmanovic.
The Serbian has now won both his encounters against Broady following the 6-4 4-6 6-4 encounter in California on Friday.
Earlier, Emma Raducanu progressed with a three-set win over former world number four Caroline Garcia and Andy Murray showed all his famed fighting qualities as he shrugged off a nightmare first set to beat Taro Daniel and lift the Scot alongside Roger Federer, Rafael Nadal and Novak Djokovic in earning 700 tour-level victories.