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In a wrestling match of a first half Scotland edged to a 6-3 lead after a penalty each for Chris Paterson and Ruaridh Jackson with Felipe Contepomi, back after overcoming a rib injury, slotting one for the Pumas.
Argentina lost experienced forwards Juan Martin Fernandez Lobbe and Rodrigo Roncero to early injuries.
Contepomi levelled it with a 63rd-minute penalty but drop goals by Jackson and his replacement Dan Parks had Scotland looking good at 12-6 until Amorosino jinked through four defenders and then watched as Contepomi slotted the vital conversion.
Both teams, and pool leaders England, are still in the hunt for the two quarter-final places which will be decided next weekend when England play Scotland and Argentina play Georgia.