Mauricio Sperduti scored the winning goal with a quarter of an hour to play, as Arsenal continued their impressive defensive record in the cup this year.
Arsenal midfielder Ivan Marcone went on a winding run that culminated in a foul just shy of the penalty area.
However, the ball flicked backwards out of the incident and the referee allowed the advantage, with Sperduti sneaking in to tuck away the winner much to the annoyance of the Godoy Cruz defence.
Arsenal have conceded just once in four cup matches this year, and that goal was a consolation away at Platense - with all three of their home ties producing clean sheets.
The Estadio del Bicentenario would have been the last place Godoy Cruz would have wanted to visit in their form, as they chalked up their fourth scoreless performance in all competitions.
Arsenal will meet All Boys in the final four.
