An inquest jury concluded last year that the marksman who killed him in Tottenham, north London, did not behave unlawfully.
The family is challenging a High Court ruling that they could not apply for judicial review.
Lord Justice Sales allowed their application for permission to appeal.
The judge warned the family against thinking his decision meant their appeal must succeed.
He said their case was "arguable" and had reached his decision on the basis that "there is a real prospect of success on appeal".
He added: "A further compelling reason for the grant of permission is that the shooting dead of a suspect by police is always a matter for careful scrutiny."
The Duggan family and their lawyers argued the jury's conclusions were contradictory.
The officer involved said he believed Mark Duggan had a weapon in his hand when he opened fire and thought he was going to shoot at police.
The jury found it was more than likely that Mark Duggan had thrown a gun onto some grass outside the car, where a weapon was found.
The three High Court judges who previously ruled out a judicial review said none of the grounds of challenge had been established.
Mark Duggan's shooting in August 2011 sparked riots in London and across England which lasted for several days.
His mother, Pamela, said she remained "deeply distressed" about her son's death and the inquest finding.
The family of Mark Duggan, who was shot dead by police in 2011, have won the right to appeal over a finding that he was lawfully killed.