Kartal, who made her national-team debut against Germany, showed composure after missing a host of opportunities in the second set.
She came back from an early break down in the first set, hitting 11 winners to Vedder’s seven as she outmuscled her in the forehand exchanges.
Kartal should have broken world number 265 Vedder in the first game of the second set, missing three break opportunities before rattling through her own service games with ease.
However, she was not helped at 4-3 up when umpire Eva Asderaki-Moore incorrectly overruled an out call on a Vedder forehand.
Had it not been overruled, Kartal would have had a break point. Instead, the point was replayed, Vedder held serve and then broke Kartal in the next game.
Vedder played her best tennis in the final two games of the second set, drawing errors out of Kartal before wrapping up the set with an ace.
The Briton put her frustration aside for the third set, quickly going a break up and putting her heavy forehand to good use to win the final five games in a row.
