Dart, the only home player to make it through the opening round, reached the quarter-finals at Edgbaston Priory last year, and she was looking for a repeat.
Dart forced two break points in the opening game, but from there, the precision game of Krejcikova began to dictate, with the second seed breaking serve in the sixth game.
A double fault in the opening game of the second set cost Dart another break, and the British No. 2 was staring at a quick defeat when Krejcikova opened up a 5-1 lead.
Dart clung on grimly, retrieving one of the breaks, but Krejcikova served out the victory at the second time of asking.
In the quarterfinals, Krejcikova will take on Russian Anastasia Potapova, while Diana Shnaider moved into the last eight with a 7-6 (7-5) 6-3 victory over eighth seed Marie Bouzkova.
Draper became the new British No 1 on Monday, having secured the first ATP Tour title of his career by beating Matteo Berrettini in the Stuttgart Open final on Sunday.
Draper will play American fifth seed Tommy Paul, a 6-3 6-4 winner over Chile’s Alejandro Tabilo, in the quarter-final.
With top seed Alcaraz now out, second seed Alex de Minaur and third seed Grigor Dimitrov are already eliminated as well, as are sixth, seventh and eighth seeds Ben Shelton, Holger Rune and Ugo Humbert.
Alcaraz had no complaints about the result as his 13-match unbeaten streak on grass came to a shuddering halt.
“I’m not too good right now. I felt like I didn’t play well, I didn’t move well,” he said.
“Of course I have to give credit to Jack. I think he played really good tennis today,”
Leave a Reply