Vince (83), who made his maiden Test half-century, and Stoneman (53) had threatened to put the tourists in a great position after captain Joe Root won the toss and elected to bat on a flat and slow Gabba wicket.
The pair put on 125 for the second wicket after Alastair Cook (2) fell early and England were in command at 127-1.
However, Pat Cummins (2-59) – playing his first Test in Australia – bowled Stoneman and grabbed the huge wicket of Root (15), while Nathan Lyon produced a brilliant piece of fielding to remove Vince, leaving England at 196-4 at stumps with Dawid Malan (28) and Moeen Ali (13) unbeaten.
Rain led to a lengthy delay prior to the start of the second session in Brisbane, where David Warner and Shaun Marsh were cleared to play for Australia despite being under injury clouds.
While there was little in the pitch early, Mitchell Starc (1-45) got one to move away slightly from Cook, who edged to Peter Handscomb at first slip and was out in the third over.
Vince and Stoneman steadied for England but they scored slowly, reaching 59-1 at lunch with few troubles.
Rain saw the start of the second session delayed by more than 90 minutes, but the tourists were in command once play resumed.
Vince brought up his half-century with a fine square drive for four and Stoneman also reached 50.
But Vince should have departed on 68, edging the impressive and unfortunate Lyon (0-40) behind, only for Tim Paine – playing in his first Test since 2010 and lauded for his glove work by Australia’s selectors after his shock inclusion – to drop the catch.
Australia were growing desperate approaching tea, and they ended the 125-run partnership between Stoneman and Vince thanks to Cummins.
The 24-year-old paceman, troubled by persistent injuries throughout his career, got one to come back in and clean bowled Stoneman for his first Test wicket on home soil.
That brought Root to the crease, and risky running would lead to Vince’s departure shortly into the final session.
Vince tried to scamper through for a single after pushing into the covers, but Lyon reacted quickly, brilliantly picking up and hitting the stumps to run out the right-hander.
Root looked in decent form – an ill-advised attempted reverse sweep aside – while Malan was tested.
But it was Root who fell, trapped in front by Cummins trying to play across the line, although Australia needed a review to get the huge scalp.
They almost had another before stumps, reviewing a not out decision after Malan was hit on the pads by Starc with the new ball, but replays showed the delivery was going down leg.