Connolly defied cramps to play the innings of his life at Sher-e-Bangla National Stadium in Dhaka on Sunday, smashing 149 off 133 balls as Australia narrowly avoided a whitewash by edging past Bangladesh's 5-274 with three deliveries to spare.
After cruising at 5-266, the Aussies suffered a 4-5 collapse - Connolly was the last wicket - to crash to 9-271, before Adam Zampa raised victory when he drove Taskin Ahmed through the covers for four.
"We all saw one of the great knocks going around - really special from Coop, what he did tonight," Inglis said.
"He kept us in the game throughout.
"I thought he was really calm the way he managed those middle overs and the partnerships throughout.
"He scored quickly, which bought us some time at the back end when we were losing wickets."Recalled left-arm seamer Shoriful Islam (6-48) was the chief destroyer for the Tigers and could have been the match-winner - and had seven scalps - had Tanzid Hasan not grassed Zampa at wide slip.
Connolly, who hit 13 fours and six sixes, raised a masterful century off 87 deliveries. The young West Australian's previous highest score was 61no in ODIs and 64 in all List-A games.
Josh Inglis (21) got the chase off to a flier at Mirpur, before he and Matt Renshaw (0) - bowled around his legs - fell in Shoriful's first over.
It was then 3-70 when Alex Carey (8) fell to a spectacular catch from Soumya Sarkar.
Marnus Labuschagne (29), Cameron Green (27) and Oliver Peake (27) all provided valuable support for Connolly in consecutive partnerships of 64, 68 and 64.
Shoriful was on a hat-trick when he got rid of Peake and Xavier Bartlett (0) in successive deliveries, before dismissing Ben Dwawshuis (4).
Connolly, who watched the carnage unfold at the other end, played on to a Mustafizur Rahman slower ball with victory in sight.
"I have that confidence in this team to chase anything down and defend anything," Connolly said.
"From the start, I always felt we were going to get near it."
Earlier, Bangladesh appeared to be in control thanks to half-centuries from Towhid Hridoy (83), Litton Das (58no) and Mosaddek Hossain (56no).
Australia made the perfect start when Soumya (2) was bowled by Bartlett (2-47) in the first over.
When Matt Renshaw (2-44) removed Tanzid (19) and stand-in captain Najmul Hossain Shanto (24), the Tigers were struggling at 3-61.
Towhid was involved in two big partnerships of 92 with Litton and 93 with Mosaddek, which threatened to take the game out of the tourists' reach had it not been for Connolly's heroics.
Bangladesh's 2-1 series win was their fourth straight in ODIs and their first ever against Australia.
The two teams will play a three-game T20 series commencing in Chittagong on Wednesday.
