Watch: Demonstrators wrestle with police at Armenia anti-PM protest
A video showed demonstrators wrestling with police as they made arrests.
Sargsyan was elected as prime minister by parliament on April 17 after previously serving as president from 2008 to early 2018.
Critics say his election is anti-democratic and have branded it a power grab.
Armenia held a constitutional referendum in 2015 which lessened the power of the president and gave greater power to the parliament.
The state-owned ArmenPress news agency said several roads in Yerevan were fully or partially closed due to Friday's protests.
Dozens of opposition protesters attempted to block government buildings in Armenia's capital to prevent Sarksyan from chairing his first cabinet meeting on Thursday.
According to local news sources, police detained nearly 20 protesters.
Earlier this week opposition leader Nikol Pashinyan announced the beginning of the nationwide "Velvet Revolution".