Argentina drafted in its talisman Juan Martin del Potro to partner Leonardo Mayer, but the pair was beaten in straight sets by the home duo of Ivan Dodig and Marin Cilic, who won 7-6 7-6 6-3.
Cilic can clinch the famous trophy for Croatia with victory over del Potro in the first reverse singles Sunday.
He put Croatia ahead in the final with a hard-fought five-set victory over Federico Delbonis in the opening singles rubber Friday before del Potro beat Ivo Karlovic in four sets to level it up.
Watching football legend Diego Maradona was delighted by that Argentina comeback, but the doubles saw Croatia take the lead again with an assured performance from a pairing which has secured crucial wins in its 2016 campaign.
Only twice since the World Group format was introduced in 1981 has a team come back from 2-1 down in the final.
A tight first set was won on a tiebreak, 7-2, with Mayer making crucial mistakes, and the first break of the match came on his service as Croatia took control of the second.
But Cilic was broken as Argentina showed some fight, only for the comeback to be halted as the second set tiebreak also went the way of Croatia.
A further break on the Mayer delivery saw any hopes of an Argentina fightback snuffed out and it was Cilic who served it out to amid delirious scenes from the home supporters.
The ATP comeback player of the year del Potro must now hope he can show the form he displayed in beating world number one Andy Murray as Argentina knocked out holder Britain in the semifinals.
Sixth-ranked Cilic, who was a teenage member of the Croatia squad which last won the Davis Cup in 2005, will be equally determined to seal victory in the showdown fourth rubber.
"We will be rooting for Marin to finish the job tomorrow," said Dodig.
Delbonis and the veteran Karlovic are scheduled to play in the second reverse singles.