Pakistan require just three wickets on the final day of the second Test to win the series, with Sri Lanka 264 short of their target.
Abid Ali became the first Pakistan player to score hundreds in his first two Tests, while Shan Masood also made a hundred on day three.
England and Australia are planning to play Tests in Pakistan in the coming years, according to Pakistan Cricket Board chairman Ehsan Mani.
Pakistan suffered two heavy defeats against Australia, but Azhar Ali claimed there were still positives to take forward.
Mitchell Starc claimed four wickets and Pat Cummins finished with three as Pakistan were bowled out for 240 at the Gabba on Thursday.
At 16 years of age, promising bowler Naseem Shah will be given a chance to show what he can do on the global stage for Pakistan.
After Sarfraz Ahmed's sacking, new Pakistan coach and chief selector Misbah-ul-Haq has named a new-look squad for the tour of Australia.
Pakistan decided to stick with Sarfraz Ahmed as captain in September but, after a dismal series defeat to Sri Lanka, he has now been sacked.
A 201-run stand between Azhar Ali and Asad Shafiq gave Pakistan a first-innings lead, but New Zealand responded to restrict them to 348.
Two early wickets from Colin de Grandhomme rocked Pakistan, but Azhar Ali and Haris Sohail dug in to make 81 apiece on day one.