Grybauskaitė has signed a relevant decree, the President's Office said on Friday.
Dambrauskienė, 46, holds a master's degree and has 17 years of experience of judicial work.
Lithuania's former prosecutor general Darius Valys stepped down in mid-June after expiry of his five-year term. Later in June, parliament rejected the appointment of Kaunas Regional Court chairman Nerijus Mačiulis to the post.
On 27 May Lithuanian government has allowed renewing more flights to Germany and the Netherlands .