The BBC's Natalia Antelava in Almaty says the outcome of the poll is easy to predict, with Mr Berdymukhamedov the favourite.
Mr Niyazov's unexpected death caused consternation because he left no designated successor.
He ruled for 21 years, tolerating no dissent as he crafted an elaborate personality cult, reduced spending on education and healthcare, and isolated his country from the outside world.
But analysts say there are some signs that the new leadership may be willing to introduce some changes.
An international think-tank says that there are hints that they may be considering relaxing the political repression exercised under Mr Niyazov.
International Crisis Group said it is not known how many people were imprisoned or disappeared under him.
But the group said that there were unconfirmed reports that two former ministers had been released from prison and that some officials were discussing whether to review the cases of political prisoners.