Finch returned to form by clubbing three sixes and five fours in his 44-ball 63, the linchpin of his side's 179-5, but the Australia captain went off in Ireland's reply with a hamstring problem.
While Australia wrapped up a 42-run win after Ireland were bowled out for 137 in 18.1 overs, there will be concerns about the severity of Finch's injury and whether it has any impact on his tournament.
He said at the post-match presentation ceremony: "I've had a bit of a hammy twinge, I don't feel too bad at the moment but generally overnight it can stiffen up a bit. I will have a scan (on Tuesday)."
If that were not bad enough for the hosts and defending champions, David did not field due to a similar issue to Finch while Stoinis, who has a history of side complaints, went off after bowling one over.
Australia, though, moved level on points with New Zealand at the top of their Super 12s group and took a sizeable wedge out of the net run-rate deficit they had accrued after their defeat to the Kiwis.
It could have been better for Australia after Ireland lost five wickets in 13 balls to slump from 18 without loss to 25 for five but Lorcan Tucker's unbeaten 71 off 48 balls limited the damage.
England will therefore leapfrog Australia into second place - only the top-two teams in each group progress to the semi-finals - if Jos Buttler's side overcome New Zealand at the Gabba on Tuesday.
Australia left-arm quick Mitchell Starc admitted they had missed an opportunity to further improve the net run-rate ahead of their final fixture against Afghanistan on Friday - England play Sri Lanka a day later - but insisted they achieved their principle objective of coming away with a win on Monday.
"I guess you could look at that," Starc said. "I mean, first and foremost, we need to win these games, so we tick that box.
"For us it's just trying to win these games because if you don't win, it doesn't matter, does it? We brought that run-rate back a bit, and we'll see how we place in 24 hours."
Ireland's chances of reaching the last four have likely vanished after their second defeat in this main stage, having qualified from the preliminary first round.
Ireland shocked England in Melbourne last week but they made a woeful start with the bat here and captain Andy Balbirnie, whose wicket was the start of a dramatic collapse, feels they missed a trick.
"It's a bit of a missed opportunity," he said. "There was naturally a lot of pressure on them, the home team. They probably have to win two games by a certain amount.
"The wicket was pretty good and there was two or three dismissals, including myself up top, that could have maybe not happened. We came here to win, and we're disappointed we didn't win.
"It's a results business, so you want to win, but you have to sometimes accept that a team who are world champions can have a decent day out and turn you over."
While Josh Little collected two for 21 and Barry McCarthy three for 29, Tucker was the standout with the bat and carted Starc for seven of his nine fours while he dispatched Josh Hazlewood for six.
"It was a brilliant innings," Balbirnie added. "The way Lorcan played when his back was against the wall, he came out throwing punches, and that's the kind of message we've given to the guys.
"Certainly the character and the fight that we showed after that start is a positive to take towards the end of the week (they finish their group campaign against New Zealand on Friday)."