Czech Krejcikova, who only returned to action last month after a back injury had kept her out of action since November, needed two hours and 40 minutes to secure the win.
In windy conditions at the Devonshire Park Lawn Tennis Club, Krejcikova committed five double faults in the opening set and twice lost serve, but built up momentum after a couple of rain-enforced breaks to overpower Dart.
Krejcikova continued to struggle with the wind in the second set and was involved in a heated exchange with the umpire after earning a time violation for taking too long on serve, with Dart taking advantage of her opponent's vulnerability.
The 124th-ranked Dart broke to level the set at 6-6 and sailed through the tiebreaker as Krejcikova's second serve faltered.
Dart also had the upper hand in the decider after breaking early and earning two match points, but the Wimbledon champion raised her level with her back against the wall and brought the set back on serve.
A deflated Dart let out a frustrated scream after losing her advantage and crumbled in her final service game as she was broken to love.
"It was a really tough match and I expected that Harriet is going to play well because she's home favourite and she loves to play on grass," said Krejcikova, who suffered a first-round exit at Queen's Club last week.
"So I expected it's going to be tough and to be honest, in that third set when I was down and having those two match points down, I was kind of already in the locker room mentally."
Krejcikova faces another Briton next in Jodie Burrage who beat Japan's Moyuka Uchijima.
British number one Emma Raducanu looked like following Dart out as she struggled against American Ann Li but the former US Open champion recovered to win 6-7(5), 6-3, 6-1.
"It was tough and there were lots of ups and downs but from the middle of the second set I managed to find an extra gear," an emotional Raducanu, who faces Australia's Maya Joint next, said.
"It feels incredible. I want to thank the crowd for getting me through some really sticky moments.
"It really meant a lot to me and I'm really proud of how I fought back after losing the first set. I'm very, very happy to come through that.
Raducanu missed last week's Berlin Open due to a back problem she has been nursing since before the French Open.
But, aside from a slip and fall on the grass in set one, she appeared to come through her opening examination on England's south coast unscathed.
Raducanu next faces Australian teenager Maya Joint, who knocked out two-time Wimbledon runner-up Ons Jabeur on Monday.
Former Australian Open champion Sofia Kenin of the United States was beaten 6-4, 6-4 by Kimberly Birrell.