When Ellis was joined at the crease by number 11 Spencer Johnson (7 from 13 balls), the tourists needed 27 runs for victory, 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.
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 148 for three, 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 Mitch 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 on his unbeaten 36.
Zimbabwe all-rounder Brad Evans was the pick of the home bowlers with 4-47 in 10 overs.