The 27-year-old, Smith suffered the injury while training with his English Championship side Ipswich Town last week and will be out for six weeks.
He's been replaced in the squad by one time international James Musa who has recently signed for Kansas City in the MLS.
"We are disappointed to lose a player of Tommy's experience and quality ahead of two very important games," said Hudson.
Hudson said Smith's injury presented a great opportunity for Musa who has played one game previously for the All Whites in 2014.
The All Whites assemble in Auckland from 27 August to begin preparing for their first game against the Solomon Islands in Auckland on September 1st.
The winner of the home and away playoff will play the fifth-ranked South American side in November for a place at the World Cup in Russia next year.