Debretsion Gebremichael, who leads the Tigray People’s Liberation Front (TPLF), also says his fighters have retaken another key city.
The government denies the claims and insists the TPLF has been crushed.
Hundreds of people have reportedly been killed in the month-long conflict.
Thousands have been displaced, while the UN and aid agencies have voiced concern about the humanitarian situation.
With all phone lines and the internet cut off, it is impossible to independently verify statements from either side.
That attack led to the start of the federal army’s operation in the region.
He also said the TPLF had retaken the city of Aksum from the federal army.
On Monday, Fisseha Asgedom, a former Ethiopian ambassador to the UN with affiliations to the TPLF, told the BBC that reports by the Addis Ababa government of a sweeping advance were “a joke”.
“The Addis Ababa government announced that they have captured Mekelle and six hours after they announced this they bombarded Mekelle… It’s propaganda. Why can’t they send the international community some videos?”
What does the Ethiopian government say?
Mr Abiy announced that Mekelle had fallen to the federal army on Saturday, describing the move as the “last phase” in his government’s offensive.
In a BBC interview on Monday, Minister of Democratisation Zadig Abraha, denied the TPLF’s claims that they had captured Aksum and shot down a plane.
He also said the TPLF were “not engaged in a guerrilla war” but were “running for their lives”.
“Their rank and file have disbanded. Everyone is surrendering,” he added.
Redwan Hussein, a senior federal official in Tigray, said government soldiers had been careful not to involve civilians in their advance.
Troops “just left the towns encircled and then moved on to the next. That’s how we see no casualty out of our operation until we also reached Mekelle,” he said.