Baltacha put in a brave effort after her surprise selection in what proved to be the deciding rubber, going down 7-5 3-6 6-1 to Maria Irigoyen.
The 29-year-old Baltacha is Britain’s number five these days, only in the squad after Heather Watson withdrew as she recovers from glandular fever.
But, despite battling to one set apiece against the world number 202, the former British number one wilted in the heat as GB lost out.
“I gave it everything and I’m obviously really disappointed," Baltacha said.
"She didn’t give me any pace and it was very hard work to keep dictating every single point. Thought I was on my way when I broke to start the third but it wasn’t meant to be.”
Robson had earlier lost 6-4 4-6 6-2 to Paula Ormaechea to leave Argentina one win away from victory.
“I’m disappointed not to win my match today and not get the team 2-1 up," Robson said.
"Ormachea played better than I was expecting and her backhand really surprised me. It was tough with the crowd but a good learning experience. I know we’ll come back next year stronger.”
Johanna Konta had been expected to face Florencia Molinero, but Baltacha stepped in in for the Australian-born world number 164, who was easily beaten by Ormaechea in the first rubber.
Argentina also made a change as Irigoyen – who is ranked higher than Molinero but did not figure in the first day’s action – came in for the vital win.
In the absence of Watson Britain had hoped Robson - the world number 38 and comfortably highest-ranked player on both sides - would win both her rubbers to give them a fighting chance of victory.
The doubles clash between the two nations is now a dead rubber.
Saturday, April 20
Johanna Konta (GBR) lost to Paula Ormaechea (ARG) 3-6 2-6
Laura Robson (GBR) beat Florencia Molinero (ARG) 6-1 6-1
Sunday, April 21
Laura Robson (GBR) lost to Paula Ormaechea (ARG) 6-4 4-6 6-2
Elena Baltacha (GBR) lost to Maria Irigoyen (ARG) 7-5 3-6 6-1