At lunch, Australia were 72 for three in 19 overs after rain stopped play for 29 minutes midway through the session.
Rogers, given the all clear following the dizzy spell at Lord's that forced him off the field on the last day of Australia's crushing 405-run win in the second Test which levelled the five-match series at 1-1, was 35 not out and Adam Voges unbeaten on 12.
Australia captain Clarke opted to bat first on winning the toss despite overcast conditions and a livelier pitch promising more assistance for the seamers.
James Anderson, England's most successful Test bowler but wicketless at Lord's on a pitch with no sideways movement, needed a mere eight balls to take his 407th Test wicket when he had David Warner, Rogers's fellow left-handed opener, lbw for two.
More From This Section
Rogers's dizzy spell was attributed to a balance problem in the inner ear caused as a result of being struck on the helmet by an Anderson bouncer during the course of his Test-best 173 in Australia's first innings at Lord's.
Rogers, however, could only watch as Finn -- whom he captained at Middlesex -- enjoyed a dramatic return to the Test arena after coming on as first-change.
The 26-year-old, playing his first Test since the 2013 Ashes opener at Trent Bridge after losing his action and being deemed "unselectable" as England suffered a 5-0 series thrashing in Australia in 2013/14, struck with the sixth ball of his first over on Wednesday.
Smith was out for seven and Australia were 18 for two in exactly eight overs.
Finn then clean bowled Clarke, now with two hundreds in his last 27 Test innings, with an excellent yorker for just 10.