Allen blasted six sixes in a tally of 62 off 42 balls as the Black Caps chased down 130-7 with ease at Hagley Oval as New Zealand claimed their second win of the series.
The hosts' victory leaves both Pakistan and New Zealand on two wins and a defeat each, with the Black Caps next facing Bangladesh on Wednesday, also in Christchurch.
The three nations are using the week-long tournament to fine tune for the T20 World Cup in Australia later this month
An improved bowling performance saw Pakistan restricted to 130 for seven in their 20 overs before Allen and Devon Conway's rapid opening partnership of 117 set up a comfortable victory.
After Allen was stumped by Mohammad Rizwan, captain Kane Williamson saw New Zealand home with three overs and five balls to spare.
"It was an improvement again from the last game," said Williamson, referring to New Zealand's eight-wicket win over Bangladesh on Sunday.
"The condition weren't easy, but we were nice and clinical with the ball.
"Finn really broke the back of the chase by being super aggressive and Devon complemented him nicely."
Following his unbeaten 70 against Bangladesh, Conway struck five fours in compiling 49 not out off 46 balls.
"We need to discuss our mistakes ahead of the next match," said Pakistan captain Babar Azam, whose side play Bangladesh on Thursday.
"We had a bad day, we just weren't up to the mark with our batting."
New Zealand bowler Michael Bracewell was named man-of-the-match after removing prolific opening pair Rizwan, for 16, and captain Babar Azam, for 21, in his superb two for 11 from four overs.
"I was a bit fortunate in that first over with a couple of short, wide ones which went straight to fielders," said Bracewell.
"It was pretty nerve-wracking, bowling to some world-class players, but I was happy with the way it came out."
Iftikhar Ahmed top scored for Pakistan with 27, including three fours.
Tim Southee removed Iftikhar and Mohammad Nawaz with the first two deliveries of the final over, but Mohammad Wasim survived the hat-trick ball.
