to help his team cash in on New Zealand's uncharacteristically sloppy fielding and post 177 for six at the Pallekele International Cricket Stadium.
Returning to defend the total, Pakistani slow bowlers did not allow the rival top order to accelerate and off-spinner Ajmal (4-30) denied them the late burst as New Zealand managed 164 for nine wickets in 20 overs.
Shahid Afridi got one for 30 and even though he did not get a wicket, Hafeez sent down four tidy overs conceding only 15 runs to stifle the opponents.
Despite the defeat, New Zealand have booked a place in the Super Eight stage because of their healthy net run-rate. Pakistan need to avoid a heavy defeat against Bangladesh in their remaining group match on Tuesday to join them in the next stage.
The most successful Twenty20 team, who have made semi-finals in all three previous editions of the tournament, Pakistan top order clicked on Sunday, aided by the poor catching of their opponents.
Hafeez could have been dismissed for a duck in the first over but New Zealand captain Ross Taylor dropped a regulation catch at slip.
Bowler Kyle Mills went through the agony again in the 11th over when Rob Nicol, fielding in the deep, not only dropped Jamshed but palmed the ball over the rope for a six.
Jamshed's 35-ball knock, which earned him the man of the match award, included four sixes and a couple of boundaries.
For New Zealand, Nicol (33) and Kane Williamson (15) added 53 runs for the opening stand but the asking run rate kept spiralling.
Even Brendon McCullum, who scored a record Twenty20 century against Bangladesh on Friday, could not break the shackles and fell to Umar Gul after a 31-ball 32.
New Zealand captain Ross Taylor scored brisk 26 runs off 11 balls but was run out in the penultimate over and Ajmal claimed two wickets in the last over to give his team a winning start in the tournament.
Brief scores: Pakistan 177/6 in 20 overs (Nasir Jamshed 56, Mohd Hafeez 43, Imran Nazir 25, Umar Akmal 23; Tim Southee 2/31, Jacob Oram 2/44), New Zealand 164/9 in 20 overs (Rob Nicol 33, Brendon McCullum 32, Ross Taylor 26; Saeed Ajmal 4/30, Shahid Afridi 1/30).