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.
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,” Hunt said.
“You get a couple of these guys and next minute your stands are full and everything becomes a lot more interesting for people.”