Cummins grabbed the wickets of Mahmudullah, Mahedi Hasan and Towhid Hridoy in consecutive deliveries over two overs as Australia's bowlers produced a disciplined performance to contain the Bangladeshis at the Sir Vivian Richards Stadium in North Sound.
Cummins, restored to the starting line-up, finished with figures of 3-29 off four overs while spinner Adam Zampa also impressed with 2-24.
Mitchell Starc meanwhile became the most prolific bowler in the history of white ball cricket with his 95th World Cup wicket after trapping Tanzid Hasan in the opening over.
Starc's early breakthrough saw him pull one clear of Sri Lanka's Lasith Malinga, setting the tone for Australia's bowlers after captain Mitchell Marsh won the toss and opted to bowl.
Australia's bowlers kept Bangladesh on the back foot after Starc's opening wicket, with restored Josh Hazlewood opening his spell with a maiden.
Another tidy over from Starc left Bangladesh with just eight runs off the first three overs, although Najmul Hossain Shanto signalled a more aggressive approach with a six off Hazlewood to start the fourth over.
Litton Das lashed a pair of boundaries off Starc in the fifth over as Bangladesh's run rate ticked up to 5.40 runs per over at 27-1.
But the introduction of Zampa slowed the Bangladesh batting, and the Aussie spinner soon had Das bowled out after tempting him to sweep in the ninth over to leave Bangladesh at 58-2.
Glenn Maxwell struck next in the 10th over, Rishad Hossain caught by Zampa at short third man for two as Bangladesh reached the halfway stage of the innings on 67-3.
Shanto's dismissal by Zampa for 41 in the 13th over slammed the brakes on for Bangladesh however, leaving them on 84-4 before Cummins ripped through the lower order to leave Australia a target of 141 to win.
