Robson, who won mixed doubles silver with Andy Murray at London 2012, completed a 3-0 win with the help of women's doubles partner Johanna Konta as they saw off Jelena Simic and Jasmina Kajtazovic.
Watson, who reached the second round at last summer's Olympics, had earlier followed Anne Keothavong's lead in winning her singles match as the Britain-Bosnia and Herzegovina clash was finally played in Israel.
Heavy rain on Wednesday forced the Europe/Africa Zone Group I tie to be delayed by 24 hours however it didn't affect Britain much in the first of their three Pool B matches.
Keothavong, who was making her 39th Fed Cup appearance to take her into second on the British all-time list for the competition, defeated Dea Herdzelas 6-4, 6-2.
Watson also won in straight sets, 6-3, 6-4 over Anita Husaric to ensure victory for the British team in Eilat before Robson and Konta then took to the court to complete it.
Only two teams will progress out of Europe/Africa Zone Group I and into the World Group II play-offs in April with Britain facing Portugal next on Friday and then Hungary on Saturday.
Should Britain top Pool B they will face the winners of Pool D, which involves Bulgaria, Luxembourg, the Netherlands and Slovenia, in an attempt to keep their promotion hopes alive.