At lunch, England were 24 for one in their second innings, eight runs behind, with captain Alastair Cook and Jonathan Trott unbeaten on 17 and five respectively after Ryan Harris clean bowled Joe Root (two) with a superb seaming delivery that clipped the top of off stump.
England, who at 2-0 up in the five-match series had already retained the Ashes following the drawn third Test at Old Trafford, dismissed Australia for 270 earlier Sunday after they added 48 runs to their overnight 222 for five.
Brad Haddin added just one to his overnight 12 before he played back to off-spinner Graeme Swann, bowling from around the wicket, and was lbw to a delivery that hit him in front of middle stump.
Haddin reviewed but, with DRS indicating neither an edge nor that the ball had pitched outside leg stump, New Zealand umpire Tony Hill's decision was upheld and England had a wicket before the new ball with Australia now 224 for six in the 76th over.
Hill originally ruled not out but England overturned his decision on review and Rogers, who batted for nearly six hours, was out having faced 225 balls with 14 boundaries.
Cook took the new ball as soon as possible, with Australia two behind at 236 for seven off 80 overs, and then, at first slip, held a low catch to dismiss Peter Siddle off James Anderson, who then had Nathan Lyon lbw.
