Robson overcame a shoulder problem to overcome world number 138 Greta Arn 0-6 6-2 6-1.
"After the first set I thought, 'this can't get any worse,'" admitted Robson. "I regrouped and played better in the second and third sets."
Watson followed it up with a routine 6-3 6-2 victory over Timea Babos, ranked 79 in the world.
“I was really happy to turn it around from yesterday and I was much stronger mentally today," Watson said.
"I didn't panic when she got back to three-all, sticking to my game plan and the second set was easier. It feels great to get through to the play-offs tomorrow and we will give it everything we have got."
Robson returned to the court with Johanna Konta for the final doubles clash against Hungary. She lost her unbeaten record for the week, however, as they fell 6-4, 2-6, 6-2 to Reka-Luca Jani and Katalin Marosi.
Coupled with victories over Bosnia-Herzegovina and Portugal in Eilat this week, GB have swept through their pool group unbeaten and with one more victory could seal promotion to World Group II.
On Sunday GB will take on Bulgaria for that place in the play-offs.