In a shadow side featuring three newcomers and without a host of top names as the World Cup looms, it was left to their all-time leading T20 wicket-taker Zampa to spearhead their push with his excellent 4-24 at the Gaddafi Stadium that did most to restrict Pakistan to 8-168 in the first of the three-match series.
But Australia's reply on Thursday proved a disappointment, and once stand-in captain Travis Head had departed for 23 after an early brisk cameo and Cameron Green had holed out on 36, they never looked like approaching their target, finishing on 8-146.
The impressive Xavier Bartlett, who'd earlier been Zampa's best support with the ball with his 2-26, did at least give it a belated go, smiting an unbeaten 34 off 25 balls, but the game had already gone as Pakistan's spinners strangled the Aussie middle-order.
It left Pakistan savouring their first victory in a T20 against Australia for nearly seven years and after seven straight defeats, which they'll see as a boost going into the World Cup in India and Sri Lanka next month.
Australia, though, looked predictably short of weaponry with captain Mitch Marsh rested and without the likes of Pat Cummins, Josh Hazlewood, Glenn Maxwell and Tim David, whose pre-Cup injury concerns left them unavailable.
Of the three debutants, Test batter Matt Renshaw got run out for 15 after a horrendous mix-up with Green, and allrounder Jack Edwards went for a wicketless 25 off his two overs and failed with the bat (five).
But at least there was some cheer for the 20-year-old Perth Scorchers quick Mahli Beardman, who after getting his first ball dispatched for six by Saim Ayub and going for 13 off his first over, recovered impressively and, handed the death over, ended up with two tail-end wickets in successive balls.
The youngster's hat-trick ball, alas, was a no-ball which was edged through the vacant slip area.
After losing the toss and being asked to field, Head, taking the reins as Marsh grabbed a break after the Scorchers' BBL triumph, couldn't have envisaged a better start than Bartlett eking out the wicket of Sahibzada Farhan with a tame caught-and-bowled on the very first ball of the innings.
Pakistan regrouped quickly though, with Ayub (40) and captain Salman Agha (39) putting on a swift 73 for the second wicket, only for Zampa to stop the rot by tempting them to both hole out in the deep in quick succession.
The leggie then struck again, trapping Babar Azam, on 24, lbw on review. The most economical of all the Aussie bowlers, Zampa then completed his fine work by snaring Usman Khan as his third victim caught at long-on. It meant he'd dismissed all four of Pakistan's top-scorers.
Player of the match Ayub got both Australian openers with his spin, bowling Matt Short for five before Head, who'd hit two sixes among the 13 balls he faced, miscued to give Babar one of his three nicely-judged catches in the deep.
The key moment came when mystery spinner Abrar Ahmed, who bowled superbly for 2-10 off his four overs, was introduced. His first over began with the run-out miscommunication between Renshaw and Green, before he then bowled Cooper Connolly for a second-ball duck.
Abrar later snaffled a struggling Josh Philippe for 12, while Mitch Owen got unlucky when he was run-out at the non-striker's end, as Mohammad Nawaz's deflection of a Green drive deflected on to the stumps.
