Scotland, who have not beaten Argentina since 1990, remain below the Pumas in the IRB rankings after this narrow reverse.
Dan Parks crossed for an individual try in the first half and added a drop goal while Chris Paterson kicked three penalties and a conversion for Hadden's side.
Edinburgh winger Francisco Leonelli touched down in the corner for the Pumas and Federico Todeschini kicked three penalties and two conversions - but it was the 70th-minute penalty try they were awarded which secured the victory for Argentina.
After Todeschini had stroked his side ahead in the seventh minute, Dan Parks brought Scotland level in the 15th minute with a well-struck drop goal.
Eight minutes later Hadden's side put together their first fluent period of play when Sean Lamont and then Hogg brilliantly weaved their way to the edge of the Argentina 22.
The ball was recycled and spun out to Parks, who chipped over the rushing defence and gathered his own kick to register his third try for Scotland, which Paterson converted.
The Edinburgh full-back edged Scotland further ahead with a penalty four minutes later - after Leonelli had tackled his Gunners team-mate Marcus Di Rollo without the ball - but Todeschini brought his side to within seven points of Scotland at the break with his second successful kick.
Paterson increased Scotland's lead by three early in the second half only for the Pumas to hit back in the 53rd minute when Leonelli shrugged off an unusually indifferent tackle from the Scotland man to brilliantly score in the corner.
Todeschini converted with no little skill from the left wing and then traded penalties with Paterson to leave the match finely balanced at 19-16 with 20 minutes left.
Props Craig Smith and Allan Jacobsen entered the fray for Scotland - as did Chris Cusiter - but Scotland were reduced to 14 men with a quarter of an hour remaining when Taylor was sin-binned for playing the ball on the ground.
Scotland were placed under intense pressure by the Argentina pack and after conceding a succession of penalties, the visitors were awarded a penalty try by referee Kelvin Deaker - which Todeschini converted - to give them victory.