The interview, for the pay-television Sky network, which is owned by Rupert Murdoch's News Corp, was conducted in Melbourne last month and broadcast in New Zealand on Tuesday.
The broadcaster had intended to screen the interview on December 25, but brought it forward after the world number one said last Friday he would be taking an "indefinite break" from golf and admitted to infidelity in his marriage.
The interview, conducted by respected sports broadcaster Murray Deaker, principally focused on Woods' relationship with his New Zealand caddie Steve Williams, although Deaker did ask Woods about his family.
"Family first and golf second. Always be like that?," Deaker asked.
"Always," Woods, who was mostly stony faced throughout the interview, replied. "It has been great actually. It has been the best thing that ever happened."
Woods said he learned family values from his mother Kultida.
"She was the disciplinarian in our family. You would think with my Dad (Earl) being a former Special Forces operative he would be the tough guy, but no, he was the softy.
"My mum was tough," he added while breaking into one of his few smiles.
Woods said since becoming a father he now had less time to devote to his game, although time management skills he learned at Stanford University, where he had to juggle study, fitness training, practice and playing, had been invaluable.
"I learned a lot at Stanford and certainly it has applied once I had a family.
"I don't practice as much as I used to. I don't spend as much hours on the golf course as I used to.
"My focus is so much more intense than it used to be, because I don't have the time.
"Whatever I have to get done, is in a shorter amount of time."
(Writing by Greg Stutchbury; Editing by Nick Mulvenney)