Paraguay had found themselves 2-0 down at half time following goals from Sergio Aguero and Lionel Messi, only for Barrios and teammate Nelson Valdez to grab second-half goals to earn Ramon Diaz's side an important point against the tournament favourites.
Argentina had the chance to take the lead in the sixth minute when Angel Di Maria lifted a cross in from the left and Nicolas Otamendi met the ball, but could only guide his effort wide of the target.
Martino's men continued to push for an opener and they almost found it through Messi, who reacted well to a shot from Javier Mascherano, only for the Barcelona man to see his effort fly just past the post.
The Argentine breakthrough came soon after, though, as Aguero made no mistake with his second major chance as the Manchester City frontman profited from a sloppy back-pass to push his side in front.
Martino's side ensured that they took a healthy lead into the interval nine minutes before the end of the half, as Messi stepped up to slot home from the spot following more poor defending that saw Samudio needlessly bring down Di Maria inside the box.
Paraguay seemed to revive themselves during the interval and they soon found a lifeline when Valdez produced a stunning long-range effort out of nowhere, with his shot nestling in the top corner from all of 30 yards out.
Argentina sought to kill off any hopes of a comeback as Messi and Di Maria both saw chances thwarted by some good last-ditch defending, before Victor Caceres gave Martino's outfit a further scare when he struck inches wide of the post from close range.
Paraguay continued to attack and they almost forced an equaliser but for the exploits of Sergio Romero in the Argentina goal, as he pulled off a fine point-blank save to tip Samudio's shot over the crossbar.
In a game that moved from end to end in quick succession, Argentina pushed for a goal to close the result off and almost found it through Marcos Rojo, but the defender could not keep his attempt down as he blazed over the target from 12 yards.
Diaz and his side turned the game on its head in the closing minutes as Argentina-born Barrios gambled from a poorly-cleared free kick to slot home from 12 yards and earn his side a vital draw.