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England trailed from the ninth minute but winger Chris Ashton crossed for the game's only try in the 78th to claim victory and Pool B in a contest dominated by kickers.
England flyhalf Jonny Wilkinson slotted a drop goal and two penalties but missed a further five attempts at goal as the 2003 champions showed little attacking ambition.
Scotland's Dan Parks, an early replacement after five minutes when flyhalf Ruaridh Jackson left the field in tears because of injury, slotted a penalty and a drop goal while fullback Chris Paterson kicked two penalties.
The Scots picked up a bonus point but are likely to miss out on the quarter-finals with Argentina expected to claim victory and second place in the group when they play the final Pool B match against Georgia on Sunday.