Argentina substitute Leonardo Mayer ended Britain's reign as Davis Cup champions with a 4-6, 6-3, 6-2, 6-4 win over Dan Evans in the fifth rubber of the semi-finals on Sunday.
Mayer, who unexpectedly replaced in-form Juan Martin del Potro for the deciding match, propelled the South American nation into their first final since 2011.
Andy Murray, whose defeat by Del Potro in a five-hour thriller in the opening singles on Friday left Britain facing an uphill task, had pulled the hosts level at 2-2 by beating Guido Pella 6-3, 6-2, 6-3 in the first of the reverse singles.
But Evans, ranked 53rd in the world, could not finish the job and Argentina secured a final showdown with Croatia.
Marin Cilic sent Croatia into the Davis Cup final with a 6-3, 6-2, 7-5 victory against Richard Gasquet on Sunday that gave the hosts a 3-1 win over France.
The world number 11 relied on his big serve and his booming forehand to down Gasquet, who had given France the first point by beating Borna Coric on Friday.
Cilic equalized on Friday when he saw off Lucas Pouille and he paired up with Ivan Dodig to give Croatia the lead with their win in the doubles against Pierre-Hugues Herbert and Nicolas Mahut on Saturday.
Croatia will play either defending champions Britain or Argentina, who were level at 2-2 in Glasgow.
"We played so many ties away, we had been waiting to play at home and now we just won in front of our home crowd it's amazing," said Croatia captain Zeljko Krajan.
"Marin was our leader all the way, he won everything despite the pressure that was on his shoulders."
After an early trade of breaks, Cilic took the measure of his opponent, who sent a backhand long to give the Croat a decisive break in the eighth game.
Cilic breezed through the second set and after he broke in the first game of the third, there was not much hope left for Gasquet.
The world number 17, however, rallied back from 2-0 down but dropped his serve in the 11th game. That left Cilic to wrap it up on his first match point.
France were without the injured Gael Monfils and Jo-Wilfried Tsonga, their two best ranked players.