Robson cruised past Dia Evtimova 6-0, 6-4 in the opening rubber of the World Group I play-off at the Municipal Tennis Club in Eilat.
But Watson had to work harder, coming through with a battling 1-6, 6-4, 6-2 success against Tsvetana Pironkova.
Britain's World Group II play-off will take place on the weekend of April 20-21, with the venue and opponents to be decided on Wednesday.
World No.45 Robson asserted her authority early against Evtimova, racing through the first set before Evtimova briefly threatened to make a match of it by breaking Robson's serve early in the second set.
However, the Brit swiftly broke back and the game then went with serve until 5-4, when Robson finished things off in under an hour.
In contrast, world No.41 Watson had a tough time against 44th-ranked Pironkova. The Bulgarian won the first set in 25 minutes before Watson took the second in a decider and the third by a safe 6-2 scoreline.
BASKETBALL: Glasgow Rocks have moved up to third place in the BBL Championship after a 94-84 victory against Plymouth Raiders.
Donald Robinson was the Rocks top scorer with 24 points.
ICE HOCKEY: The Braehead Clan remain sixth in the Elite Ice Hockey league after going down 2-0 away to Coventry Blaze.
ATHLETICS: Hurdles specialist Eilidh Child has smashed the Scottish 400 metres record, clocking 52.06s to win her semi-final at the Sheffield based UK Championships.