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Dart broke first but went on to lose eight games in a row to fall a set and a break down.
However, with three breaks of her own the world No 173 forced a decider against her Japanese opponent.
A typically tight and chaotic third set followed, with both players breaking three times before it went to a tie-breaker.
And though Dart trailed 5-1, she bounced back to nine of the next 10 points to win the tie-break 10-6.
The 2-6 6-4 7-6(6) victory means Dart is through to face fourth seed Simona Halep in the second round.
Another late finish in Melbourne saw fellow Briton Cameron Norrie fall to Pierre-Hugues Herbert in a marathon five-set encounter.
Norrie trailed 1-0 before going 2-1 up in sets, but ultimately lost 7-5 3-6 3-6 7-5 6-4 as the Frenchman roared back.
Highlights of Edmund's defeat to Lajovic
Svitolina beats Boulter as Edmund, Konta also lose
Fifth seed Elina Svitolina battled into the Australian Open second round as Katie Boulter joined the list of first-round British exits on Tuesday.
Svitolina needed 81 minutes to dismiss Boulter 6-4 7-5 on the 1573 Arena in Melbourne.
Svitolina is targeting a maiden Grand Slam final having reached the semis at Wimbledon and the US Open last year.
Highlights of Konta's crushing defeat to Jabeur
The Ukrainian faces a potential third-round clash with Garbine Muguruza, but must first overcome Lauren Davis in the second round.
On Tuesday, Svitolina was made to work as Boulter held her own early on, but the Brit was broken in the 10th game to go a set down.
A topsy-turvy second set saw Boulter break first, only for Svitolina to move 5-2 up.
Svitolina failed to serve out for the match at 5-3, but eventually saw off Boulter with a break at 6-5.
'I'm happy to be playing again' - Konta points to "small victories" after first round exit