The reigning champions led twice through Lionel Messi (29 minutes) and Lisandro Martinez (93, extra time) only for the World Cup debutants to level twice - first via Deroy Duarte's low drive from the angle (59) and then an incredible goal from Sidny Lopes Cabral curling into the top corner for 2-2 in extra time.
Cuti struck the decisive blow in the second half of extra time (111) when he met Messi's corner with a towering header that took a deflection off Diney Borges and nestled in the far corner.
Marcos Senesi remained an unused substitute as Argentina set up a Round of 16 clash against Mo Salah's Egypt in Atlanta on Tuesday evening (5pm).
Djed Spence and Pedro Porro return to action in the Round of 16 on Monday - England take on Mexico at the Azteca (1am) before Pedro's Spain face Portugal in Dallas (8pm).