The explosions occurred seconds apart outside the capital's main train station as hundreds gathered for the rally, organised by Turkey's public workers' union and other civic society groups.
The rally aimed to call for increased democracy and an end to the renewed violence between Kurdish rebels and Turkish security forces.
There was no immediate claim of responsibility for the deadliest attack in Turkey in years.
At a news conference, he declared a three-day official mourning period for the victims as well as for all those killed in terror attacks since July.
"For some time, we have been receiving intelligence information based from some (Kurdish rebel) and Daesh statements that certain suicide attackers would be sent to Turkey... And that through these attackers chaos would be created in Turkey," Davutoglu told reporters, using the Arabic acronym for the Islamic State group.
The Turkish government imposed a temporary news blackout covering images that showed the moment of the blasts, gruesome or bloody pictures or "images that create a feeling of panic." A spokesman warned media organisations they could face a "full blackout" if they did not comply.
Many people in Ankara reported being unable to access Twitter and other social media websites after the blasts. It was not clear if authorities had blocked access to the websites, but Turkey often does impose blackouts following attacks.
Authorities had been on alert after Turkey agreed to take a more active role in the US-led battle against the Islamic State group.
Turkey opened up its bases to US aircraft to launch air raids on the extremist group in Syria and carried out a limited number of strikes on the group itself.
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