Watson and Robson, who both competed at the London 2012 Olympics, the latter winning mixed doubles silver with Andy Murray, both contributed as Britain topped their pool in Israel.
Britain beat Bosnia and Herzegovina 3-0 and Portugal and Hungary both 2-1 to set up a clash in the Europe/Africa Zone Group I promotional play-offs against Bulgaria, who they brushed aside.
Robson began the winning performance against Bulgaria by seeing off Dia Evtimova, ranked 208 places below her in 253rd in the world, in straight sets 6-0, 6-4.
And then Watson completed it against more testing opposition in Tsvetana Pirnokova, who is ranked just three places behind her in the world in 44th, 1-6, 6-4, 6-2.
Britain join Poland in progressing through to the World Group II play-offs in April where they will hope to fare better than they did last year.
Sweden proved too much of a force in the 2012 World Group II play-offs as they ran out 4-1 winners against Elena Baltacha, Anne Keothavong, Robson and Watson.