The 24-year-old was edged out in a fiercely-contested opening set but levelled things up against her Czech opponent with a show of defiance in the second.
However, Allertova’s superior world ranking eventually told as she claimed a 7-6 (19/17) 4-6 6-2 victory on Court Five.
The pair could not be separated in the opener, forcing a tie-break which saw world number 144 Konta take a 5-2 lead and then set herself up with two set points, which were quickly wiped out.
That was merely the beginning of the drama, though, as the players saved set point after set point before Allertova finally hit the winner that counted.
Konta came back strongly in the second set and, after seeing two break points saved in game four, she claimed the first break of the match two games later.
Allertova hit back immediately but the Briton claimed the decisive break in game 10 to take the match into a decider.
At that point, however, the Czech turned up the pressure and Konta was unable to find any answers. She was broken twice more by the world number 86, who then served out for the match despite some more late resistance from her opponent.