Warner, transformed from the man who struggled in Sunday's defeat by India, busied his way to 107.
With captain Aaron Finch also clubbing 82, Australia should have posted more than their 307 all out, only to lose their last six wickets for 30 runs thanks mainly to Mohammad Amir's 5-30.
Facing a more reachable target than they would have once feared, Pakistan were well placed at 136-2 when their own collapse of 4-24 sucked the life from the chase.
Hassan Ali hammered 32 from 15 balls and Wahab Riaz 45 from 39 balls to revive their vociferous fans, but after Wahab was given out caught behind on review off Mitchell Starc, Pakistan were bowled out for 266.
Pakistan stay eighth in the 10-team table and move on to play arch-rivals India at Old Trafford on Sunday knowing that another defeat would go a long way to ending their involvement in the tournament.
Australia join New Zealand at the top on six points, having played a game more than the Kiwis.
Not only was it a relief for cricket to return after two days of abandonments, it returned with a magnificent contest that fluctuated throughout.
Despite the grey and chilly conditions, this was a wonderful spectacle. The intimate setting is unique for a tournament mainly being played at large arenas and the vocal spectators filled the boutique venue with noise.
The match itself - played on a juicy surface that offered encouragement for the pace bowlers - was rarely lacking in entertainment.
The early Australia charge was helped by some poor bowling and fielding, with Pakistan's resurgence first led by Amir and followed up by a third-wicket stand of 80 between Imam-ul-Haq and Mohammad Hafeez.
After Imam gloved Pat Cummins behind, the Pakistan implosion left the crowd silent, only for raucous hope to return through the hitting of Hassan and Wahab.
Just as Australia were getting desperate, Starc asked for a review against a hopping Wahab. It detected the faintest edge and, eight balls later, the game was over.
Even without the hullabaloo of the hostility that Warner's return to international cricket has attracted, the left-handed opener is having an eventful tournament.
On Saturday, he was left shaken when a stroke he played in practice struck net bowler Jaykishan Plaha on the head and, the next day, his ponderous 56 from 84 balls against India drew criticism.
Before this game, he was backed by Finch and responded with a characteristic performance of punchy strokes, bustling running and the occasional display of power.
Finch, dropped twice, hit four leg-side sixes in their opening stand of 146 before offering a leading edge to Amir.
Warner took over. When he edged Shaheen Afridi between keeper and slip, it took him to three figures and sparked an emotional celebration of his trademark leap and prolonged gestures towards the Australia dressing room.
He was dropped on 104 but holed out soon after, part of an overall slide of eight wickets for 84 runs, wickets falling regularly because of Australia's inability to kick on and Pakistan's improvement.