Although Alex Danson's late close-range strike set up a grandstand finish, it proved to be little consolation.
To Team GB's credit, they were the better side after the break but the South Americans, who are looking to win a golden treble by adding the Olympic Games title to their world and Champions Trophy crowns, were just too experienced to let a two-goal lead slip.
Argentina, now in their second Olympic Games final, after Sydney 2000, are looking to improve on the bronze medals they won in 2004 and 2008 and now face a rematch of the 2010 World Cup final against the Netherlands.
They controlled most of the first half thanks to a high press which gave the hosts no outlet once they had possession.
From there, they were able to build pressure but what did that more effectively was Barrionuevo's sixth-minute penalty corner strike, which she flicked low through the legs of defensive runner Crista Cullen to beat goalkeeper Beth Storry.
Luciana Aymar's run along the baseline caused more problems but Storry saved her initial shot and when the ball looped up the Argentina captain ambitiously hooked over.
GB failed to make Florencia Mutio make a save in the first half, with Danson's reverse-stick shot into the sideboard of the goal the closest they got.
But the right side of defence was an avenue Argentina sought to exploit frequently and when Aymar burst down the line, taking the ball close to being out of play, she set up Rebecchi for another run along the baseline.
Storry came to clear but took the player and not the ball, yet somehow Rebecchi maintained her composure to get it back under control and force it over the line from a narrow angle.
Cullen's penalty corner early in the second half was bravely charged down by Macarena Rodriguez Perez, Ashleigh Ball hit a shot well wide and Danson forced the first save out of Mutio as the hosts stepped up the tempo.
Argentina just sat back and defended and invited the home side to break them down, and even when they did, which was not often, they failed to take the opportunity with Helen Richardson mis-hitting a shot off target when she was found in space.
Richardson made amends with five minutes to go with a cross from which Danson could not miss, but it came too late and the home team could not force extra-time.