Godman's Edinburgh team-mates Hugo Southwell and winger Simon Webster are also drafted in, alongside Glasgow flanker John Barclay.
Hadden has also rung some tactical changes following Saturday's 21-15 defeat in Rosario.
Ally Hogg will move to number eight to make way for openside flanker Barclay, while Chris Paterson will make his record 88th Scotland appearance on the wing, with Southwell replacing him at full-back.
Wingers Simon Danielli and Thom Evans drop out of the 22, alongside Glasgow number eight Johnnie Beattie.
Hadden said: "The only reason for the changes is to finish the season with a win.
"Phil has been playing consistently well for some time now and deserves this opportunity. It makes sense if we are to get the best out of him to surround him with familiar faces.
"John Barclay's inclusion reflects our need to perform better at the breakdown."
Mike Blair will becomes the 25th Scot to reach the landmark of 50 caps in the Buenos Aires clash, while Nick De Luca returns to the 22 for the first time since the RBS 6 Nations loss to Ireland.