Watson got the ball rolling with a straight sets victory over Jelena Ostapenko, the British number three edging a tight opening set 7-5 before blowing her opponent away 6-1 in the second.
Konta then took on Diana Marcinkevica and having taken the opening set 6-3 she then lost the second 6-4 and found herself 5-2 down in the decider.
But she reeled off five consecutive games to take it 7-5 and give Britain their first victory in this year's competition.
Britain are just one of 17 nations competing in Budapest for a play-off tie in the Fed Cup World Group II, with the first phase being a round-robin format where Judy Murray's team have been drawn in Group B.
The team, which consists of Watson, Konta, Tara Moore and Jocelyn Rae, will next face Romania tomorrow followed by hosts Hungary on Sunday.
� Sportsbeat 2014
- Sports & Recreation
- Diana Marcinkevica