The overwhelming victory ensured New Zealand remain in Group I with the prospect of challenging for a World Group playoff next year.
New Zealand took an unassailable 3-0 lead when they won the doubles on Saturday and finished off Pakistan with straight-set victories in the reverse singles on the final day.
Tearney beat Muhammad Abid Ali Khan Akbar 6-1, 6-3 and Statham downed Pakistan number two Samir Iftikhar 6-4, 6-1.
Davis Cup: Pakistan drop back to Group II
New Zealand captain Alistair Hunt said getting a World Group qualifying tie for the first time in nearly 18 years would be huge for New Zealand.
"Imagine getting a world group team over here like (the) States or even Australia, it doesn't really matter. All these guys are unbelievable and top 20-50 players," said Hunt.
"You get a couple of these guys and next minute your stands are full and everything becomes a lot more interesting for people."
New Zealand lead Pakistan in Davis Cup tie
Overall results
Finn Tearney (NZL) bt Muhammad Abid (PAK) 6-1, 6-3
Jose Statham (NZL) bt Samir Iftikhar (PAK) 6-4, 6-1
Michael Venus/Marcus Daniell (NZL) bt Aqeel Khan/Muhammad Abid (PAK) 6-0, 6-1, 6-2
Jose Statham (NZL) bt Muhammad Abid (PAK) 6-0, 6-1, 6-4
Finn Tearney (NZL) bt Samir Iftikhar (PAK) 6-2, 6-0, 6-0
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