Other protesters pushed into the election commission offices to demand that they resign over voting irregularities and fraud.
"The demonstrators are acting like hooligans and violating social order," Sainbayar, a police spokesman, said.
The Democratic Party, led by Tsakhia Elbegdorj, has accused the MPRP of stealing Sunday's vote.
|Extra police in riot gear were rushed to the ruling party headquarters [EPA]
The MPRP, a former communist party, says that it won 45 seats in the 76-seat Great Khural, while the Democrats have reportedly won 21 seats.
The General Election Committee has yet to make a formal announcement on the ballot.
"We do not accept these results," Elbegdorj told a news conference earlier on Tuesday.
"No one needs these kinds of results, and they will be corrected in accordance with law."
International observers say that overall the election was free and fair, but new election rules that changed the first-past-the-post system to one of multi-member constituencies have led to procedural problems and confusion.
'Free and fair'
Bayar called for calm in a televised address from inside the party headquarters before the fires was started.
"The other party [the Democrats] is accusing us of buying the election, it's not true, the election was free and fair. We now request that everyone should stop this chaotic protest immediately," Bayar said on local Eagle Television.
"Elbegdorj made a false announcement and he is misleading people and inciting violence."
A prolonged dispute about the election results could further delay the ratification of long-awaited agreements that would allow production to go ahead on a massive copper and gold project.