The Premier League club agreed a record fee with St Pat's for a League of Ireland player in order to sign Melia on January 1, 2026.
Melia, who turns 18 in September, emerged as a Celtic target last month with club chiefs thought to be keeping tabs on the exciting prospect.
However, Spurs were in pole position to land the forward despite initially having a £1m bid thrown out by the St Pat's.
Postecoglou's side reached terms with St Pat's in the final days of the transfer window to see Melia pen a five-year deal beginning in January next year. The striker will remain with St Pat's until the transfer goes through.
Melia is the youngest-ever player and goalscorer for St Pats' having broken his own record twice as a 15-year-old.
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He has scored 10 goals for the Irish side in 54 outings and received a first Republic of Ireland under-21s call-up in October.