Pakistan started off their chase of 148 with a slow pace, finding themselves in trouble at 44-4 after losing Mohammad Rizwan for 16 and Salman Agha for a golden duck. Johnson's three-wicket haul wreaked havoc on their top order.
Usman Khan hit his first half-century after being dropped by Matthew Short but fell for a defiant 52 as Johnson struck again in his final spell.
Johnson then removed Abbas Afridi for four to finish with 5-26 before Irfan Khan launched another counter-attack with an unbeaten 37 but he could not take the tourists over the line despite only 16 required from the final over.
Pakistan were dismissed for 134 when Haris Rauf was run out.
They had recovered from an early onslaught by Australia's opening batsmen to restrict the hosts to 147-9 despite several dropped catches as Rauf claimed 4-22 and Abbas Afridi bagged 3-17.
Jake Fraser-McGurk and Short hit five boundaries and three sixes between them in the first 15 balls of the match at the Sydney Cricket Ground as Australia reached 50 runs in only 3.1 overs after electing to bat first.
Pakistan hit back through Rauf, who had Fraser-McGurk caught for 20 while he attempted a cross-batted hit and then got rid of skipper Josh Inglis for a duck two deliveries later thanks to a diving catch by Sufiyan Muqeem.
Short missed a slower one from Abbas Afridi to fall for 32 and Marcus Stoinis was dismissed by Muqeem for 14 after he had twice survived due to some sloppy catching by Salman Agha and Shaheen Afridi.
Glenn Maxwell shook off a top-edge that rattled his helmet grill and looked to accelerate but he holed out against Muqeem for 21 to leave Australia in a bit of trouble at 95-5.
Another dropped catch, this time by Babar Azam, handed the dangerous Tim David a lifeline and he carted Rauf for two fours in the 15th over before becoming the seamer's third victim.
Rauf then castled Xavier Bartlett for his 107th T20I wicket to join Shadab Khan atop Pakistan's all-time list.
Pakistan won the preceding one-day international series 2-1 before a 29-run defeat in the rain-shortened T20 series opener.
The third and final match takes place in Hobart on Monday.
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