* Grybauskaite broadly backs existing austerity measures

* President can influence economic policy, veto budget laws

* She says some tax rises a mistake, wants to push exports (Adds 96 percent of votes counted)

By Nerijus Adomaitis

VILNIUS, May 18 (Reuters) - European Union budget commissioner Dalia Grybauskaite was elected as the first woman president of Lithuanian, the Baltic nation battling a deep recession, official results showed on Sunday.

The 53-year-old, a tough-talking former finance minister, ran as an independent candidate, enhancing her popularity in contrast to the main political parties, whose standing were hit by the economic downturn and allegations of corruption.

With 96 percent of votes counted, she secured a little over 68 percent support. Turnout was 51 percent, just above the 50 percent needed to give her an outright first round win.

The country, an EU and NATO member since 2004, experienced its worst post-Soviet rioting in January.

In Lithuania, the president is the head of state and formally appoints the prime minister and the cabinet.