Krejcikova, a two-time major winner in singles, had previously served for the match at 6-5 up in the second set.
Czech Krejcikova has played just five matches this year after a lengthy lay-off with a back injury.
She lost her opening grass-court match of the season at Queen’s last week but the battling nature of her win over Dart will give her confidence before her Wimbledon title defence.
Krejcikova will face another British player in Jodie Burrage in the second round.
Burrage, who missed last year’s Wimbledon with an ankle injury, beat Japan’s Moyuka Uchijima 6-3 6-2.
Britons Emma Raducanu, Sonay Kartal and Mimi Xu are also in action on the second day in Eastbourne.