The victory also marked the British qualifier’s first win against a top 20 player.
The 12th-seeded Ukrainian broke the Briton four times in the opening set at the BNP Paribas Open before Dart steadied herself in the second.
O britânico number three, ranked 122nd, managed to break Svitolina three times consecutively to lead 5-3 then smashed a crucial ace to claim the set.
Dart rushed out to a 3-0 lead in the decider but Svitolina scraped back some points before the match – which took two hours and 17 minutes – finished 2-6 6-3 6-3 in the qualifier’s favour.
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, Enquanto isso, 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.
Mais cedo, 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.