The arrests came after police and protesters clashed during an opposition rally over disputed election results.
The EU said it was "appalled" and demanded Mr Kozulin and other detainees be freed immediately.
Belarus' interior ministry said 52 people were arrested on Saturday, but human rights groups say the true figure was much higher.
One local civil rights group said as many as 600 people have been arrested in Belarus during the last week.
Belarusian Interior Minister Vladimir Naumov said Mr Kozulin had been arrested because he had called for people to rise up against the state.
Witnesses said Mr Kozulin - who was a runner-up in the presidential election - and several members of his family were pulled from the crowd by police.
"We threatened to go with a big column of journalists and supporters and stand outside the remand centre. Only then did they admit they were holding him there," Mr Kozulin's wife Irina said.
The main opposition candidate at the election, Alexander Milinkevich, attempted to visit Mr Kozulin on Sunday but when he rang at the entrance to the Zhodino centre he was turned away by a metallic voice telling him to come back on Monday.
Mr Milinkevich promised to launch an information campaign that would counter the authorities' claims that the opposition had resorted to violence.
A stepson of Mr Milinkevich is among about 250 opposition supporters said by human rights groups to be held in Zhodino in connection with Saturday's clash with police and with the days-long protest on Minsk's October Square.
Most of those who camped out on the square were handed administrative sentences of up to two weeks for hooliganism.
Political analyst Sergei Pankovsky said the Belarussian opposition was still struggling to find its response to growing public discontent.
"What's happening is the search for a way to use this popular discontent. Even under conditions of an information blockade, people came out onto the streets.... The opposition only aligned itself with them," said Mr Pankovsky.
Mr Kozulin's arrest unleashed a new chorus of protest from the West at the Belarussian leadership's tactics.
"The United States deplores today's use of force by Belarussian authorities against peaceful demonstrators in Minsk," said US State Department spokesman Sean McCormack in a statement.
But State television in Minsk showed a series of "vox pop" interviews with members of the public who unanimously defended the election, saying they had voted for the stability they believed Mr Lukashenko had brought.