The surprise move came just days before the Museum of Fine Arts in Bern is expected to reveal whether it accepts the inheritance of the spectacular trove of more than 1,000 pieces amassed during the Nazi era.
Uta Werner, 86, a cousin, appealed to a court in Munich to be recognized as Gurlitt's heir, with the backing of her children and some other family members, the spokesman said in a statement.
A report by a psychiatrist that "seriously questioned" Gurlitt's mental fitness to make a will had prompted the family to now act, it said.
Gurlitt, who died in May aged 81, had hoarded more than 1,000 paintings, drawings and sketches, including masterpieces by the likes of Picasso and Chagall, in his Munich flat for decades.
Hundreds more works were unearthed at his Salzburg home.
He was the son of Nazi-era art dealer Hildebrand Gurlitt, who was tasked with selling works taken or bought under duress from Jewish families, and avant-garde art seized from German museums that the Hitler regime deemed "degenerate".
Before he died, Cornelius Gurlitt struck a deal with the German government to help track down the rightful owners of the artwork.
A day after his death, the Museum of Fine Arts in Bern said it had been astonished to learn that it was named as the recipient of his collection in his will.
It is to announce Monday at a news conference in Berlin whether it will accept the bequest.
The spokesman for Uta Werner said the family had set out plans last week which foresaw the "unconditional" return of any looted artworks and transparent provenance work, if the court backed their motion.