Captain Finch's 63 off 44 balls helped the champions to an imposing total of 179-5 at the Gabba before Australia's bowlers led by Mitchell Starc terrorized Ireland's top order on Halloween.
Australia move to second in Group 1 behind leading New Zealand, who have a game in hand, but the hosts missed a golden opportunity to lift their net run rate above third-placed England after reducing Ireland to 25 for five early on.
A brave, unbeaten 71 by Lorcan Tucker rallied Ireland to 137 before they were bowled out after 18.1 overs.
Tucker was dropped on 35 by Pat Cummins, leaving bowler Starc fuming.
"We didn't get it in the right area enough (to Tucker)," Starc said.
"We take that catch and we could be sitting here talking about how we snuck back in the net run rate or whatever."
Nonetheless, with the top two teams advancing to the semi-finals, the pressure is on Jos Buttler's England to take down unbeaten New Zealand at the same venue on Tuesday.
Australia have other injury worries apart from Finch, who spent most of Ireland's chase off the field.
Number six batsman Tim David was also spared fielding duties with hamstring soreness.
All-rounder Marcus Stoinis was on and off the field despite scoring 35 and taking a wicket during his one over of bowling.
Finch said he hoped to be fit for Australia's last Super 12 clash against Afghanistan in Adelaide on Friday but the skipper has a history of hamstring problems.
"That doesn't feel too bad at the moment but generally overnight they can stiffen up," said the stocky opener.
The fitness concerns soured victory for the hosts in front of a crowd laden with fans in Halloween costumes.
After a slow and awkward 31 not out in the win over Sri Lanka in Perth, Finch started tensely with the bat but grew in confidence in partnerships with Mitchell Marsh (28) and Stoinis.
He brought up his fifty with a six during a horror over for Ireland all-rounder Mark Adair, who bowled five wides and conceded 26 runs before finishing with ugly figures of 0-59.
Ireland's Barry McCarthy had a better night with 3-29 and leaped over the rope for a spectacular catch which denied Stoinis a six and saved Adair four runs when he threw the ball back in before landing.
Left-armer Starc was not called into action until the fourth over of Ireland's chase but made an immediate impact, bowling Curtis Campher and George Dockrell for ducks in a thrilling burst of pace and swing.
That reduced Ireland to 25 for five but it was to be the apex for Starc and Australia as Tucker dragged them into a grind that could ultimately prove costly if run rates are decisive.
