The 16-year-old from Waikato has been training at the club since the start of the month while the formalities went through.
Woods is a highly-coveted young forward in New Zealand and has agreed a two-year deal with Albion, which has already seen him working with fellow Academy recruits and the wider first-team squad.
His progress will be eagerly followed in his homeland where his father, Grant, revealed his son had been adjusting well to his new life.
He said: "He's a laid-back character but he loves team training and he's settling in well."
National under-17 coach Colin Tuaa believes Woods has the ability to succeed and provide a significant role model for future New Zealand footballers. He said: "He's a big lad but he's got great feet, a natural nose for goal and a good temperament. I'm confident he'll do well."