The British number four dug deep to secure the biggest win of her career in three sets against Beatriz Haddad Maia.
After taking the first set on a tie-break, Dart looked in trouble when Brazilian Haddad Maia took the match into a decider.
The 22-year-old played some confident, high-class tennis, but when her levels dropped and she became frustrated Haddad Maia took advantage.
However, Haddad Maia needed extensive treatment on a hip problem and a refocused Dart reeled off five games in a row to run out a 7-6 (4) 3-6 6-1 winner.
Serving at 5-4, world number 182 Dart was two points from clinching the first set but Haddad Maia broke back and then held to take the lead for the first time.
A bold service game from Dart took the set into a tie-break and, back in aggressive mode, she took it impressively.
Dart, whose ankle was already strapped, required further treatment during the changeover and on the resumption promptly dropped serve.
They exchanged breaks again, Dart slipping from 40-0 up, and Haddad Maia went on to level the match.
At 1-1 in the decider Haddad Maia took the medical time-out and Dart, sensing her chance, stepped on the gas to book a dream third-round showdown with Barty.