The visitors, set a 281 victory target, began a rain-delayed day on 107 for three but their chances looked remote when they were reduced to 227-8.
But Cummins, in partnership with Nathan Lyon, launched a fightback and sealed victory with an unbeaten 44.
In suffocating tension, he struck the final boundary to take his side over the line for a 1-0 lead in the series.
After a rain-delayed start, unflappable Australia opening batsman Usman Khawaja had kept his team on course for victory with a gritty innings to edge the visitors towards the target.
They reached 183-5 at the tea interval — requiring 98 more but a match full of momentum shifts 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 were closing in on victory. But Cummins sparked the fightback with huge sixes off Root’s off-spin as England delayed taking the new ball and was the calmest person in the stadium as he secured victory.
It was the second-highest successful victory chase at Edgbaston and made up for Australia’s heart-breaking loss to England on the same ground in 2005 when they fell three short of chasing down 282.
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