Buses temporarily closed off the Brankov bridge - the main connection between the old and new parts of Belgrade.
Their biggest success is a walkout at the country's two largest coalmines, which is already affecting power stations and threatens to cause blackouts as coal stocks are used up.
The Russian President, Vladimir Putin, has offered to intervene in the crisis, by meeting both candidates in the coming days, but Moscow indicated that it expects the second round to go ahead on Sunday.
Back to normal
In Belgrade, rubbish lorries, trolleybuses and trams blocked major intersections, and several hundred opposition supporters formed a human barricade on one of the city's main avenues.
Buses temporarily closed off the Brankov bridge - the main connection between the old and new parts of Belgrade.
People were forced to walk to work in strong wind and driving rain, but by late morning the city was reported to be back to normal.
Opposition leaders had hoped to paralyse the country.
They have called for a campaign of civil disobedience to climax on Wednesday, but correspondents have questioned whether their action will be solid enough to force the authorities to back down.
Election officials say Mr Kostunica did not receive enough votes to win outright in the first round of voting.
The opposition says Mr Kostunica won more than 50% of the vote on 24 September, and have promised a boycott of the second round.
Coal crisis looms
After thousands of miners went on strike in Kolubara and Kostolac over the weekend, three of six generators at a power station producing half the country's electricity have been closed down.
The thermal plant at Kostolac is reported to be running at one fifth of its capacity.
President Milosevic has said he will not resort to force to make miners return to work, though special police sealed off one of the Kolubara pits on Sunday.
The BBC's correspondent in the region, Paul Wood, says the police were pulled back after several thousand local people arrived to support the miners.
Russian offer
President Putin's offer to mediate between Mr Milosevic and Mr Kostunica is a blow to the opposition, because the Kremlin refers to both as "candidates" for the "second round".
Statements by German Chancellor Gerhard Schroeder had earlier said that Mr Putin agreed that Mr Kostunica's "election victory" reflected Yugoslavs' wish for democratic change.
At a news conference in Belgrade, Mr Kostunica said Russian policy had been indecisive.
Earlier the Russian Foreign Ministry said Mr Milosevic had declined an earlier Russian offer of mediation by Foreign Minister Igor Ivanov.
Elswhere in Yugoslavia, the main north-south railway was blocked by protesters at Pozega, and a major road junction was blocked at Cacak, severing the main road linking Belgrade with the south of the country.
Traffic was also disrupted in Mr Milosevic's home town of Pozarevac, and in the eastern town of Bor.
There were reports of demonstrations in Serbia's third largest town, Nis, and the western town of Uzice.
Eight local radio stations controlled by the ruling Socialists have announced that they have stopped transmitting the official government broadcaster's programmes.