Polling stations opened across Turkmenistan at 03:00 GMT and were to close at 13:00 GMT with preliminary results expected by Monday and the official outcome on Wednesday.
Many Turkmen express hopes that Berdymukhammedov, 49, will put an end to the totalitarian Niyazov era and open up the nation to the outside world.
Niyazov, who died last December, ruled Turkmenistan for 21 years, overseeing a powerful personality cult, allowing only one political party, and tightly controlling the media.
Statues of the Niyazov are everywhere, the most famous a shiny gold colossus that rotates to face the sun.
A woman casts her ballot
in Ashgabat [Reuters]
Students are also forced to study the writings of Niyazov, who renamed the month of April in honour of his mother and January after himself.
Berdymukhammedov was a long-time ally of the former president, but in the run-up to Sunday's election he broke new ground by expressing support for limited reforms, including an eventual end to the one-party system.
In another first for the gas-rich ex-Soviet republic, a small team from the Organisation for Security and Co-operation in Europe (OSCE) were present for the elections, although not as official observers.
Analysts say this shows that Berdymukhammedov wants, at a minimum, to change Turkmenistan's profile.
"It's a question of international image. He needs a formally contested election," one Western diplomat based in Turkmenistan said.
On Saturday, a diplomat from the OSCE also said that there were already signs of an opening up of the famously closed state.
"The answer to us of course was very positive: Yes we want to engage, yes we... want to be participating," Goran Lennmarker, the OSCE parliamentary assembly president, said.
Some believe that change will inevitably follow.
"Niyazov's successor can't keep the same control," Eduard Poletayev, a political analyst from Kazakhstan said. "Some liberalisation needs to happen."
But others point to the depth of Niyazov's legacy and warn that change is unlikely to be quick.
"Turkmen society hopes for change, but is not ready yet," Atyum Ulunian, a historian at the Russian academy of sciences in Moscow said.