Scotland looked to have kicked their way into the quarter-finals when they moved into a 12-6 lead.
However, having suffered similar heartbreak against England, it was the Pumas who snatched victory with seven minutes left through the only try of the match.
Argentina had had plenty of chances to put the game to bed before the late drama, but once again Martin Rodriguez and Felipe Contepomi struggled with the boot.
However Contepomi landed a crucial eight points to leave Scotland quarter-final hopes hanging by a thread.
Scotland will now have to beat England in their final game to stand a chance of reaching the last eight.
Ruaridh Jackson had weighed in with six points while Chris Paterson landed a penalty and Dan Parks came on to slot over a drop-goal but it wasn't quite enough.
A breathless opening to the encounter saw Paterson miss an early shot at goal, while Rodríguez was wayward with a drop-goal attempt.
Contepomi's 17th-minute penalty also dropped short, with the Pumas unable to profit from their work at the breakdown.
However the Argentina skipper made no mistake two minutes later as Scotland were again penalised at the ruck following Santiago Fernandez's break.
Rodríguez saw his penalty effort from distance sail wide but Scotland made them pay as first Paterson and then Jackson landed penalties in the final three minutes to give them a 6-3 lead.
Scotland came agonisingly close to a try on 48 minutes when Evans chipped over the top and won the race to the ball and although John Barclay and Rory Lawson continued the drive, the Pumas were able to snaffle the ball at the breakdown.
Paterson saw his drop-goal effort go wide while Contepomi failed with another shot at goal as both sides slugged it out in worsening conditions.
Contepomi finally levelled the scores on 63 minutes with a penalty, only for Jackson to respond with a drop-goal within two minutes.
Argentina should have been level on 67 minutes, but his penalty hit the post - with Scotland opening up a six-point gap with a drop-goal from Dan Parks.
However the Pumas took the lead for the first time when Contepomi converted Amorosino's try, with the replacement winger dancing through the Scotland defence to score.
Amorosino, though, almost went from hero to villain as he spilled Parks' kick to the corner with two minutes left. From the resulting lineout, Scotland drove to the line but Parks was wayward with his drop-goal effort.
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