Marcus Stoinis, who made his ODI debut against England in September, will replace Faulkner whose injury comes six weeks before Australia begin their campaign for an elusive Twenty20 World Cup title in India.
"James reported increased stiffness in his right hamstring during the first innings tonight and found it difficult running during our batting innings," Cricket Australia's sports science and sport medicine manager Alex Kountouris said in a statement.
"Unfortunately, with the short turnaround between games we do not believe he will recover in time to take any further part in the series."
Faulkner, with 36, was Australia's second-highest scorer as they crumbled to a 159-run defeat at Eden Park. Wellington is the venue for the next match on Saturday.