Finn failed both times to claim a hat trick but was outstanding throughout and was ably assisted by James Anderson's 2-34 after England won the toss and chose to bowl at Riverside Ground.
Batting conditions were treacherous, but David Hussey top-scored with 70 from 73 balls and nine fours to help Australia recover from 6-2 and 96-6.
Australia captain Michael Clarke was the next highest scorer with 43.
After England had limited Australia to six runs from the first five overs, Finn claimed two wickets from the first two balls of the next over.
David Warner was initially given not out by umpire Nigel Llong, but a referral showed he was plumb lbw and the decision was reversed with the batsman out for 2 from 18 balls.
Finn's next ball trapped Peter Forrest lbw for a golden duck. Clarke only just survived the hat trick, the ball only missing his bat by millimeters on its way to wicketkeeper Criag Kieswetter's gloves.
Eoin Morgan dropped an easy catch off Bresnan at gully when Clarke was on 8, but Australia's batsmen were stifled until 14 came off the 14th over, Watson hitting a six and a four off Stuart Broad.
Watson (28) and Clarke put on 51 but that stand ended when Watson chopped a ball from Bresnan onto his stumps in the 19th over. Three overs later, George Bailey was clean bowled by Ravi Bopara for 9.
Bopara was denied a second wicket when he hit Hussey on the toe in the same over. The referral suggested the ball would have clipped leg stump by fractionally less than the margin needed to overturn Llong's decision.
That meant England was unable to refer a strong lbw shout in the next over from Broad against Clarke, who was also dropped by Anderson at first slip in the 25th over from an edge off Broad.
Clarke's luck finally ran out in the 33rd over when Finn clattered his off stump.
Finn then had Matthew Wade brilliantly caught behind by Kieswetter for a golden duck, but Hussey and Brett Lee mounted a patient recovery.
Hussey was dropped off Bopara when Ian Bell failed to hold a difficult chance in the 42nd, and he reached his 50 from 62 balls with a glance for two off Samit Patel.
Lee flicked Anderson for six through square leg in the 46th over, but he was caught for 27 off the next ball by Broad at long on—giving Anderson his 500th international wicket.
In his next over, Anderson removed Clint McKay for 6 when Bresnan took a steepling catch at mid off.
Hussey then fell to Bresnan in the final over, caught on the long-off boundary by Anderson, but James Pattinson dragged Australia to 200 with two from the last ball of the innings.
England leads 2-0 in the five-match series.