Presidential guard troops have been trying to disarm Mr Bemba's forces
The Dutch and German peacekeepers were on standby in nearby Gabon in case of violence during last month's elections.
Forces loyal to the two candidates in October's presidential run-off have now engaged in three days of clashes.
Both the UN and Roman Catholic Church have called for an immediate end to the fighting, and urged both sides to talk.
The archbishop of Kinshasa urged "political parties to stop sending out messages of hate that are poisoning the situation".
HAVE YOUR SAY Kabila and Bemba should both understand that its time for the people to speak
Dan, Kinshasa
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Supporters of Jean-Pierre Bemba say President Joseph Kabila's guards are attacking Mr Bemba's house.
The streets of the city centre are virtually empty and shops are closed, as people are too afraid to venture outside, but the BBC's Said Penda says the city is now calm.
Spanish EU troops and members of the UN peacekeeping force - the world's largest - are already on the streets of Kinshasa trying to maintain order.
The BBC's Karen Allen says the EU troops' first task will be to secure the international airport, which has closed.
DR CONGO RESULTS BY PROVINCE NATIONALLY: Joseph Kabila: 45% Jean-Pierre Bemba: 20% Antoine Gizenga: 13% Nzanga Mobutu: 5% Oscar Kashala: 4% Turnout: 70% Source: CEI
Violence in pictures Fighting mars peaceful poll
UN spokesman in DR Congo Kemal Saiki told BBC News it was too early to say whether the fighting called into question the second round of the elections.
"The fighting could be contained, or it could expand," he said.
Mr Bemba's spokesman Germain Kabinga told the BBC's Focus on Africa programme that the opposition leader was still committed to the run-off but that the UN should create a buffer zone between the two camps.
"We really want to go to the second round. We don't want to restart the war," he said.
On Monday, a group of 14 foreign ambassadors had to be rescued by UN and EU peacekeepers from Mr Bemba's residence on the bank of the River Congo, where they had gone to try to ease tensions and organise a meeting between the two election rivals.
UN officials say that on Monday, Mr Kabila's presidential guards tried to disarm Mr Bemba's forces following the clashes which began on Sunday, shortly before the election results were due to be announced.
Mr Bemba is a former rebel leader and retains his own personal security force but is now under UN protection.
Sunday's clashes left at least five people dead but there have been no details of subsequent casualties.
Fraud claims
Mr Kabila took 45% of the vote, just short of the 50% needed for outright victory, while Mr Bemba gained 20%.
Dead bodies still lay on the street on Monday
Some of his rivals, including those from Mr Bemba's party, say there was widespread fraud in the elections.
The 30 July election was the first democratic poll to be held in the country since it gained independence in 1960 and follows the official end of a five-year conflict, which dragged in several other African countries and led to the death of more than 3m people.
The results show a regional division in DR Congo, a country two-thirds the size of western Europe.
Mr Bemba won most votes in the west of the country, while Mr Kabila gained most support in the Swahili-speaking east.
Vote result map with Kabila's support is in red, Bemba's in blue, Gizenga's in green.
There have been three days of violent clashes in Kinshasa between supporters of Jean-Pierre Bemba and incumbent Democratic Republic of the Congo President Joseph Kabila following the release of results from the first round of Congo's elections. The two presidential candidates will face each other in a run off election October 29th after the first round of votes, July 30th, failed to produce a victor. The results were announced on Sunday with Kabila received 45% of the vote, short of the absolute majority required to avoid a run-off, while Bemba gained 20% and Antoine Gizenga won 13%. Bemba's support is concentrated in the western provinces of the country while Kabila's support is concentrated in the east.  

Bemba's supporters claim that Kabila's presidential guard is attacking Bemba's house. Most downtown streets are deserted and shops closed as people stay in their homes to avoid the sporadic fighting which left five people dead on Sunday. 

Kabila's rivals, including Bemba, claim there was widespread fraud in the elections. 

400 European Union peacekeepers from the Netherlands and Germany currently stationed in neighbouring Gabon are flying into the capital to quell the fighting. Spanish EU troops and 17,000 United Nations peacekeepers are already in the capital. 

The June 30 election was the first democratic vote to be held in the Congo since it gained independence in 1960 and occurs following the official end of five years of conflict.