After opting to bat, Australia recovered from early setbacks to post 196 for 5, thanks largely to Renshaw's unbeaten 89 off 52 balls.
The left-hander struck four fours and five sixes and anchored the innings after the visitors had slumped to 44 for 3 inside six overs.
Bangladesh had made the perfect start with Nasum Ahmed dismissing Josh Inglis for 11 before Nahid Rana removed the in-form Cooper Connolly for just one. Mustafizur Rahman then accounted for Australian captain Mitchell Marsh, who made 20.
However, Renshaw rebuilt the innings alongside Tim David, who blasted 45 from 26 deliveries, including four sixes. Their 97-run partnership for the fourth wicket shifted the momentum decisively in Australia's favour before Renshaw guided the visitors to a daunting total with support from Joel Davies (13*).
Chasing 197, Bangladesh responded positively and stayed in the hunt throughout the innings. Tanzid Hasan provided an explosive start, hammering 30 off 15 balls, while Saif Hassan continued his impressive form with 42 from 33 deliveries.
After Soumya Sarkar chipped in with a brisk 15, Parvez Hossain Emon kept Bangladesh's hopes alive with an aggressive 36 off 22 balls, sharing important partnerships in the middle overs.
Stand-in captain Towhid Hridoy then launched a late assault, smashing 35 off 22 balls with three boundaries and two sixes to bring the equation within reach. But Bangladesh struggled to find the finishing touch in the closing overs.
Australia's bowlers held their nerve under pressure. Aaron Hardie struck twice, dismissing both Emon and Hridoy at crucial moments, while Nathan Ellis, Joel Davies, Adam Zampa, and Renshaw claimed one wicket apiece.
Bangladesh needed eight runs from the final over and remained in contention until the very last ball. However, Hridoy fell attempting a big hit off Hardie on the final delivery as the hosts finished on 189 for 6, seven runs short of the target.
For his unbeaten 89 and economical spell of 1 for 13, Renshaw was named Player of the Match, helping Australia secure the series with one game still to play.
