The 25-year-old opener smashed a maiden ODI century and finished with 149 from 134 deliveries, striking 13 fours and six sixes as Australia chased down 275 with three balls to spare. His brilliance ultimately proved enough to overcome a career-best six-wicket haul from Bangladesh seamer Shoriful Islam, who almost inspired an unlikely comeback victory for the hosts.
Australia appeared to be cruising towards their target when Connolly launched three consecutive sixes off Taskin Ahmed in the 45th over, leaving the tourists needing just nine runs from the final five overs with five wickets in hand.
What followed was a dramatic twist.
Shoriful ripped through the lower order, dismissing Oliver Peake and Xavier Bartlett with successive deliveries before removing Ben Dwarshuis to leave Australia suddenly wobbling. Connolly found himself stranded at the non-striker's end for three overs as wickets tumbled around him, with Bangladesh sensing an extraordinary victory.
The turning point came when Tanzid Hasan spilled a straightforward chance to dismiss Adam Zampa off Shoriful. The missed opportunity proved costly.
Connolly eventually regained the strike but was bowled by Mustafizur Rahman with Australia still four runs short of victory. The opener remained crouched in disappointment after his dismissal, having carried the chase almost single-handedly.
With one wicket remaining, Zampa held his nerve and settled the contest in the final over, driving Taskin through the covers for four to seal a dramatic Australian win and deny Bangladesh a 3-0 sweep.
Earlier in the chase, Australia had made a positive start before Shoriful struck twice in the same over. Captain Josh Inglis, after hitting four boundaries in a brisk 21, was caught at short square leg before Matt Renshaw was bowled around his legs for a duck.
Alex Carey then departed for eight after Soumya Sarkar produced a stunning diving catch, leaving Australia in trouble at 70-3.
Connolly responded superbly. He shared successive partnerships of 64 with Marnus Labuschagne, 68 with Cameron Green and another 64 with Peake to rebuild the innings. The left-hander reached his first ODI century in just 87 balls and continued to dominate the scoring as wickets fell around him.
Bangladesh had earlier posted 274-5 after choosing to bat first, with Towhid Hridoy leading the effort with an entertaining 83 from 88 balls.
The hosts suffered an early setback when Soumya Sarkar was bowled by Xavier Bartlett with the second ball of the match, but Tanzid Hasan and stand-in captain Najmul Hossain Shanto steadied the innings before Matt Renshaw removed both batters.
Hridoy then anchored Bangladesh's recovery. He shared a 92-run partnership with Litton Das and added another 93 alongside Mosaddek Hossain as the home side rebuilt effectively on a slow surface.
Litton retired hurt with cramps during his innings but later returned to finish unbeaten on 58, while Mosaddek continued his impressive comeback with an unbeaten 56. Hridoy's aggressive knock included eight boundaries before he was dismissed in the closing overs.
Although Bangladesh struggled to accelerate at the death, adding only 32 runs in the final five overs, their total looked enough for victory when Australia's lower order began to crumble.
Instead, Connolly's extraordinary innings ensured Australia escaped with a consolation win, while Bangladesh still celebrated a historic 2-1 series triumph — their first-ever ODI series victory over Australia and a fourth consecutive series win in the format.
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