Australia were in a hurry to finish the game with openers Warner and Travis Head (31) going hammer and tongs from the very first ball.
The pair raced to 60/0 before rain interrupted play. Once play resumed, the Aussies lost a bit of momentum as Rishad Hossain (2/23 in 3 overs) got into the act.
The young leg-spinner removed Head and Mitchell Marsh (1) in quick succession.
Despite the twin blows however, Australia never looked in trouble as Warner went about his business.
Playing his last T20 World Cup, he brought up his fifty with a majestic six.
When rain stopped play for the second time, Australia were 100 for two in 11.2 overs, 28 runs ahead of the DLS par score of 72.
Earlier, left-arm pacer Mitchell Starc expectedly gave Australia the early breakthrough as he got Tanzid Hasan in the first over to become the leading wicket-taker in World Cups (95), moving past Sri Lankan legend Lasith Malinga.
Liton Das (16) and skipper Najmul Hossain Shanto (41) repaired the innings by stitching a 58-run stand.
Shanto smacked Josh Hazlewood for a six over long on in the fourth over. He then took on Starc, hitting a couple of fours in the fifth over.
But when Zampa had the ball he immediately put an end to the partnership, trapping Das leg before wicket in the ninth over.
Australian spinners Zampa and Glenn Maxwell tightened the noose around Bangladesh who could only muster 26 runs from the ninth to the 13th over while losing Rishad Hossain (2), and the key wicket of Shanto, who was dismissed by Zampa.
Hridoy (40 off 28) took Bangladesh past the 100-run mark and tried hard to inflate his side’s total. He smashed back-to-back sixes off Marcus Stoinis.
But any chance of a strong finish by Bangladesh was dashed by Cummins who made a significant impact at the fag end of the innings.
The pacer took wickets off consecutive balls in the end of the 18th over. Mahmudullah attempted a pull shot but ended up playing the ball back on to his stumps. Cummins then had Mahedi Hasan caught by Zampa.
The pacer returned in the last over to send Hridoy packing with his first ball to complete the hat-trick.