Chigumbura's side beat their opponents for the first time since 1983 after an excellent bowling effort restricted them to 209-9.
At 156-7, Zimbabwe's chances looked dashed, but Chigumbura (52 not out) joined forces with Prosper Utseya (30 not out) to carry the hosts over the line.
They won with two overs to spare, ending a run of 28 one-day losses against Australia, and a delighted Chigumbura felt the win was exactly what they deserved.
"The most important thing was to stay calm under pressure," Chigumbura said.
"Prosper did a fantastic job at the end there. He could try and take all the risks and [I had to] make sure I will be there at the end.
"The guys have worked today. Everyone has worked very hard for this. Our batting has been letting us down. We showed some improvement."
Utseya smashed Mitchell Starc (2-41 off 10) for six over mid-wicket to bring up Zimbabwe's famous win and the all-rounder also chipped in with two wickets.
"I am a bit tired but think it's a deserved win. I'm loving the moment," Utseya said afterwards.
"[It is] very pleasing. We've really worked hard.
"I said to Elton I was going to look to play more positive. Normally people tend to have fielders in for the new batter."
Zimbabwe batsman Brendan Taylor has played in five of his nation's losses to Australia and was elated that his side had finally broken their drought.
"It's only sinking in now," Taylor said.
"What a fantastic feeling. I'm over the moon.
"I wasn't born yet [when we last beat them]. Still three years to go after that!
"We've been in some good positions against South Africa and sort of chucked them away so it's great to get over the line [today]."
The win also ended Zimbabwe's seven-match losing streak in one-day international cricket.