Unflappable Australia opening batsman Usman Khawaja had earlier kept his team on course for victory with a gritty innings to edge the visitors towards the target.
But in a barn-storming match full of momentum shifts, the pendulum swung England's way when Khawaja was bowled by England skipper Ben Stokes for 65.
When Joe Root held a caught and bowled chance to remove Alex Carey, England seemed to be close to victory.
But Cummins refused to buckle, hitting Root's off-spin for two sixes in one over as England delayed taking the new ball.
England had Australia in a straight-jacket with Khawaja and Travis Head unable to score.
Persons:
Australia's Usman Khawaja, Cameron, Paul Childs BIRMINGHAM, Pat Cummins, Cummins, Nathan Lyon, Harry Brook, Usman Khawaja, Khawaja, Ben Stokes, Joe Root, Alex Carey, Root's, Lyon, Stokes, Scott Bolan, Broad, Steve Smith, Jonny Bairstow, Travis, Moeen Ali, Root, Cameron Green, Green, Ollie Robinson, Carey, Martyn Herman, Christian Radnedge, Ken Ferris, Toby Davis
Organizations:
England, Edgbaston, Cummins, Australia, Thomson
Locations:
Edgbaston, Birmingham, Britain, England, Australia, Lord's, Lyon