When Ellis was joined at the crease by number 11 Spencer Johnson (seven from 13 balls), the tourists needed 27 runs to reach their victory target of 272, something the pair managed at a run-a-ball to take their side to the win.
Zimbabwe will rue Wessly Madhevere dropping Ellis in the penultimate over when he could have sealed victory for them, as he then failed to defend nine runs in the final over when Ellis hit a six off the first ball.
Australia, who rested several of their regulars for the tour, claimed the series 3-0 and now move on to South Africa for three ODI fixtures that begin in Durban on Thursday.
"Great game of cricket," Australia captain Mitchell Marsh said.
"I thought Zimbabwe probably deserved to win but the boys got the job done at the end there and that was fantastic to see.
"We're aware we need to keep improving as a team. I thought the way we played over the three matches, albeit today we got lucky, but there's lots of positive signs with our group. We've got some young talent which is exciting for the future."
After the home side elected to bat first, Zimbabwe's innings was anchored by an unbeaten 81 from 72 balls by captain Sikandar Raza, to go with half-centuries for Craig Ervine (59 from 79 balls) and Innocent Kaia (55 from 66 balls).
Kaia retired hurt with a hamstring injury, with the score on 3-148, as he could no longer run between the wickets.
Seamer Ellis was also the pick of the Australian bowlers with 3-50 in his 10 overs.
Australia were always up with the run rate in their reply but when openers Travis Head (6) and Marsh (8) fell early, it set the tone for an innings where they lost wickets at regular intervals.
But Inglis scored a superb 103 from 102 balls as he struck 11 fours, with the next highest scorer Ellis unbeaten on 36, who battled on with Johnson, who was not out seven.
"To do it for your country and with my mate Spence at the other end there, it was pretty cool," Ellis said.
"It was just a matter of staying in there and taking it as deep as we can, which we managed to do.
"We probably let it get a little bit too close for our liking, but the way it shaped up with who had overs left, I thought if we could get through the quicks, I knew there was some spin to come. Thankfully it worked out."
Zimbabwe all-rounder Brad Evans was the pick of the home bowlers with 4-47 in 10 overs.
"I thought 280-290 was par with the bat so to make a game out of 272 was a great effort," Raza said.
"As a captain I'm really proud. Plenty to learn. I'd rather have a game like that than for the Aussies to just chase it down in 40 overs."
