The British No.1 held her nerve at 6-3 down in the opening tiebreak of her match with Dalila Jakupovic to come back, win the set and then the match 7-6 (9-7) 6-2.
GB had got off to a winning start after Katie Boulter eased past Kaja Juvan 6-4 6-2.
Konta’s victory then gave Anne Keothavong’s side an unassailable 2-0 lead before the doubles pair of Harriet Dart and Katie Swan made it a 3-0 whitewash at the University of Bath.
Boulter had got Great Britain off to a perfect beginning with a relatively comfortable victory over Juvan.
The 22-year-old hit her opponent with a barrage of punishing groundstrokes before she eventually took the first set with a single break.
And the second set was even more comprehensive with the World No.83 hitting four aces on her way to wrapping it up 6-2, finishing the match in just over an hour.
"Being here is quite tremendous. Everyone showed so much support out there," said Boulter, who was making her singles debut in the Fed Cup.
"I had a great team around me to help me with those nerves and put me in the right place mentally."
Things were not quite as straight forward for Konta, who was matched all the way by Jakupovic.
The first set took almost an hour and 20 minutes in what was a rollercoaster for Konta as she led, and then was forced to save three set points in the tiebreak.
Eventually she held her nerve to win 9-7, having come from 6-3 down, and was roared all the way by the 3,000 fans in Bath.
The second set was more straight forward as Jakupovic appeared to run out of steam and the British No.1 celebrated a 7-6 (9-7) 6-2 triumph.
Konta said: "What I tried to draw on is vast experiences - it's not the first time I've been down in a tiebreak or down in a set.
"I tried to stay calm because there was every reason for me to be calm. I thought I was competing well. I just tried to trust that more than anything else."
Captain Anne Keothavong added: "Jo did a great job to come through that first set. It wasn't easy but Jo is the ultimate pro."
That result gave Great Britain a 2-0 lead and a first Fed Cup victory on home soil in 26 years.
But they were not done there as Dart and Swan completed a memorable day by storming to a 6-2 6-2 win over the tiring pair of Jakupovic and Juvan.
Sportsbeat 2019