"We feel that Leon would not have considered machine gun fire and open ground as obstacles when a man is down,'' said Corporal Smiths' family in a written statement.
"He would have preferred to act, and risk it all, than consider the thought of his mate suffering alone.''
Lance Corporal Smith was killed in action on September 28 during an operation at a compound suspected of housing Taleban bomb-makers in Wardak province, south-west of Kabul.
He was on his second tour of duty in Afghanistan.
Commanding officer Colonel Chris Parsons spoke at Lance Corporal Smith's funeral of a determined man who braved gunfire and ran, "quick as a cat'' to be reunited with his fallen comrade.
The Chief of the Defence Force, Lieutenant General Rhys Jones, said he met Lance Corporal Smith just weeks before he was killed.
"He was a tough guy, both mentally and physically,'' said General Jones.
Lance Corporal Smith, 33, is survived by his mother, grandmother and two brothers in Wellington, and his father and grandparents in Tauranga.
The Charles Upham Award is made annually to a person deemed to have performed the most outstanding act of heroism at risk to their own life.
Nominations are called from the Defence Force, police, fire service, Civil Defence and the public.
Lance Corporal Smith is the 24th person to receive the award and the fourth member of the New Zealand Defence Force.
The SAS were deployed in Afghanistan for two-and-a-half years, returning in April.