Robson thrashed Molinero in straight sets after earlier in the day Johanna Konta lost 6-3 6-2 to Paula Ormaechea in Buenos Aires.
"I was very pleased with how I played," Robson said afterwards. "I stuck to my game plan and hitting some higher balls to her backhand and apart from a few double faults nothing to complain about. It’s 1-1 going into tomorrow, it’s going to be tough but we can totally do this."
Konta, who became eligible to represent Britain last year, had been entrusted with the opening match for Judy Murray’s team, but the world number 164 was no match for her Argentine opponent on the clay.
After a breezy start Ormaechea bagged two breaks of serve and a 3-0 lead in the opener, and closed out with confidence.
In the second set Konta had opportunities to break early on, but Ormaechea, who had won seven of nine Fed Cup singles matches on clay courts, came through confidently to wrap up the contest.
“It was a great experience being out there today and a real honour to represent GB," said Konta. "I didn’t get off to the best of starts but found my rhythm after the first four games and played better after that. I really hope Laura can level up and we can win the tie tomorrow.”
The tie now moves to a packed schedule on Sunday with Robson in doubles action alongside Anne Keothavong before the reverse singles rubbers.
"We are probably where we expected to be," Murray reflected. "We’re looking forward to tomorrow, it’s all to play for."
Saturday, April 20
Johanna Konta (GBR) lost to Paula Ormachea (ARG) 3-6 2-6
Laura Robson (GBR) beat Florencia Molinero (ARG) 6-1 6-1
Sunday, April 21
Anne Keothavong/Laura Robson (GBR) v Mailen Auroux/Maria Irigoyen (ARG)
Laura Robson (GBR) v Paula Ormachea (ARG)
Johanna Konta (GBR) v Florencia Molinero (ARG)