Neesham grabbed 3 for 25 and got the key wicket of Shahid Afridi while Mills polished off the tail with 3 for 26 to bowl out Pakistan for 127 with seven balls to spare.
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New Zealand captain Kane Williamson, who lost his second successive toss, top scored with 33 and Luke Ronchi hit a quick-fire 31 off 19 balls to lift New Zealand to 144 for 8.
“We had a review about (our) performance last night and it was nice to improve,” said Williamson, who lost the first match by seven wickets on Thursday. “Looks like the tracks aren’t free flowing and it’s important how you shape your innings.”
Afridi gave Pakistan a glimmer of hope by hitting three sixes and a four in his 28 off just 11 balls before he got a thin edge of Neesham’s short pitched delivery and was caught behind with 12 balls to go.
Neesham also got the wickets of Saad Nasim and Umar Akmal who put on the innings’ best stand of 40 runs for the fourth wicket
Mills – one of the four changes made by the Kiwis after losing the first match by seven wickets on Thursday – chipped in with the last three wickets to finish with career-best figures.
Anton Devcich’s left-arm spin with the new ball provided New Zealand the lucky breakthrough when he had Sarfraz Ahmed trapped leg before wicket off the fourth ball. Television replays clearly showed that the ball would have had missed the leg stump by a wide margin.
Devcich dealt another blow in his second over by taking a two-handed low return catch to dismiss Mohammad Hafeez as Pakistan’s batsmen struggled to score freely against New Zealand’s spinners on the slow wicket.
“I think we lost two quick wickets and then we played lots of dots balls, otherwise it was not that sort of a total not to be chased,” Afridi said.
Fit-again opening batsman Ahmed Shehzad could hit just one boundary in top scoring with 33 but consumed 36 balls before he was bowled by Daniel Vettori in the 15th over, playing his first Twetny20 in two years.
Earlier, Devcich (20) gave New Zealand a reasonable start of 49 off 42 balls before the Kiwis lost four wickets for 29 runs to Pakistan’s spinners and slipped to 78-4.
Ronchi’s rapid 31 off 19 balls that included two successive sixes off Afridi’s one over provided New Zealand a decent total as Pakistan conceded 51 runs in the last five overs.
Fast bowler Umar Gul returned from his eight-month layoff due to a knee injury to finish with 2 for 24 while Afridi bagged 2 for 33.
The teams now compete in a five-match one-day series which starts at the same venue from Monday.