They argue the Good Friday Agreement and peace process means there must be a Parliamentary vote if Northern Ireland is to leave the EU.
A High Court judge rejected arguments the case should be dealt with along with legal actions in the UK.
Northern Ireland voted to remain in the EU by a 56% majority in June's referendum vote.
The group, including Alliance leader David Ford, SDLP leader Colum Eastwood, Sinn FÃ©in MLA John O'Dowd and Steven Agnew, leader of the Green Party, are seeking to prevent the British government from formally triggering Brexit.
Raymond McCord, a victims campaigner whose son Raymond McCord Jr was killed by the UVF in north Belfast in 1997, is also mounting a bid to stop Brexit.
The judge said that issues specific to Northern Ireland could "fall between the cracks" during challenges due to come before a London court.
His decision clears the way for the politicians to make a fuller case at a two-day hearing in Belfast next week.
It will examine the group's claim that the Good Friday Agreement overrides any entitlement to quit the EU without a Parliamentary vote.
Mr McCord also contends Brexit will inflict damage on the Northern Ireland peace process and is concerned that money from the European Union, which goes towards victims of the Troubles, may be discontinued.
Government lawyers argue that the prime minister can use historic Royal Prerogative powers to start the process of withdrawing from the EU.
Mr McCord welcomed the judge's ruling.
"The judge wants to see that the Northern Irish issues are dealt with and the voice of the people is listened to," he said.
"Brexit is like marriage whereby if the wife wants to sell the house, she requires the husband's consent.
"What I mean by this is that the British prime minister is saying that 'Brexit means Brexit' but what she has to understand is that she requires the consent of the people of Northern Ireland."
A cross-party group of politicians in Northern Ireland have won the right to mount a legal challenge over Brexit.