Mr Cowan opened his eyes and called young nephews by name on Saturday. By Sunday morning, he was asking young relatives questions about rugby, he said.
"[He's asking] inquisitive questions, which made us go wow, he's very alert and he's very aware of where he is.''
He spoke in a slightly nasal voice because of medical tubes in his nose, Mr Greening said.
"Right now, he is awake and alert when he talks to us but he still needs to rest at the same time, he's come off some pretty heavy drugs that induced the coma.''
Mr Cowan remains in a serious but stable condition, but was no longer in a critical condition. The progress was "absolutely'' a relief, Mr Greening said.
Doctors have still not discussed any time frames for how long Mr Cowan was expected to remain in hospital, Mr Greening said.
"They're still looking to figure out what happened, why. We don't know answers yet. All they know is he's getting stronger, and whatever they're doing, it's helping him.''
Mr Greening said his brother-in-law was a "fine specimen'' of a man, and said his strength and fitness were likely behind his recovery.
A contingent of immediate and extended family members, sometimes up to 40 strong, have packed out the intensive care unit since he fell ill, Mr Greening said. The whanau will continue to wait at the hospital in support, he said.
Mr Greening said his family was grateful for the help, food and shelter the army had provided during the last few weeks.
The recruit fell ill during an exercise in the Hunua Ranges near the SAS base, where Mr Cowan and fellow recruits had to run an 8km course in uniform while carrying equipment in a load-carrying vest weighing about 10kg.
The group were reaching the end of the activity when he stopped and fell to the ground and became unconscious.
He received immediate first aid assistance and an army medic started treatment within five minutes of his collapse.
The incident has sparked an inquiry into selection processes for the SAS.
