Konta reeled off five games on the spin to comeback from 5-2 in the deciding set and beat Diana Marcinkevica 6-3 4-6 7-5.
Earlier, Watson had beaten Latvian teenager Jelena Ostapenko in straight sets.
A topsy-turvy opening set saw the pair exchange numerous breaks with Watson eventually edging it, before the British number two settled and blitzing her way through the second to win 7-5 6-1.
Britain is one of 17 nations competing in Budapest for a play-off tie in the Fed Cup World Group II.
The GB team consists of Watson, Konta, Tara Moore and Jocelyn Rae after Laura Robson withdrew with a wrist injury.
The first phase is a round-robin format – Britain and Latvia joined by Hungary and Romania in Group B – with only the group winners progressing to the next round.