Asif took 4-41 to reduce the Black Caps to 147-8, a lead of 244, after the hosts had dismissed Pakistan for 332.
New Zealand’s innings started poorly as Martin Guptill and Daniel Flynn were out for ducks to leave them 0-2.
Ross Taylor steadied things with 59 but Peter Fulton inexplicably failed to call for a review to overturn his lbw.
Fulton was the third Black Cap to go without scoring, after being adjudged leg before to Umar Gul.
But he did not challenge the umpire’s decision despite knowing that he had got an inside edge – a fact that was backed up by replays.
The series is being played under the Decision Review System that allows contentious decisions to be referred to the third umpire, who will use TV replays to analyse whether or not they were correct.
“He was unsure whether he’d hit the ball,” said New Zealand paceman Chris Martin.
“It was a very fine edge and sometimes you think you hit your pad or your boot or the ground, it’s hard to know.
“It’s a new format and such a new part of the game that, like most things, it’s going to take a while to get used to it and to get better at it.
“I think next time if you’ve got a hunch you’ll pretty much go for it (the review).”
Mohammad Aamer accounted for Guptill and Flynn in his first two overs as New Zealand’s second innings got off to the worst possible start.
But Taylor, who scored his second half century of the match, and Tim McIntosh put on 68 for the third wicket before the former ran himself out trying for a quick single.
His dismissal sparked a collapse as Fulton, McIntosh and Brendon McCullum followed in quick succession.
Asif trapped McIntosh leg before for 31 and had McCullum caught behind by Kamran Akmal without scoring.
That left the home side 115-6 when rain forced the players off for an early tea.
Asif produced another aggressive spell after the break and had Daniel Vettori caught by Fawad Alam for eight, the ball after he had been dropped by Umar Akmal.
Shane Bond then played on after shouldering arms to hand Asif his fourth wicket.
Earlier, Bond, making his Test comeback at the age of 34, took one of the two remaining Pakistan wickets to complete his fifth five-wicket Test haul as New Zealand mopped up the tourists’ tail inside the first 30 minutes.
“I think we may have let an opportunity slip today by not really batting them out of the game,” Martin admitted.
“But I still think with the lead that we had (97 runs on the first innings) and the runs that we’ve got now the game is poised and it’s definitely there for us to take.
“Anything over 250 on a wearing wicket with variable bounce is going to be a big ask for them, especially if we do everything in a disciplined manner and turn up prepared to win the game.
“I think we learnt quite a lot from watching them bowl on it today. They put us under real pressure by bowling real straight and making us make a lot of decisions and giving us very limited opportunities to score.
“I think if we were a thoughtful side, our bowling line-up in particular, we could take a lot from that and take it into tomorrow.”
Pakistan wicketkeeper Kamran Akmal was unsure about which way he thought the match would pan out on the final day.
“At the moment I think it’s 50-50,” he said. “This wicket is behaving totally different to the first two days. There is more uneven bounce.”
