Like Boulter Konta had to face two tie breaks as she beat Anna Bondar 6-2, 6-7(1), 7-6(4).
Konta started rapidly as she broke her young opponent in the first game of the match and eventually took the opening set 6-2.
Bondar dug deep in the second set and she held on three long service games to force a tie break.
There she was able to completely upset Konta's rhythm and easily forced a deciding set.
Both players were broken once in the final set but this time in the tie break it was Konta who came through at the University of Bath.
The victory now sees Great Britain face Serbia on Saturday and the winner of that match will then go into the Fed Cup World Group II play-off in April.
Bondar has never even played a main tour qualifier event before yet she pushed Konta hard and although she was visibly distraught at the end of the match she was given a rapturous ovation by the crowd.
Konta paid tribute to her opponent afterwards saying: "The reason why I was on the ropes was because of the way she was playing. She played amazing tennis.
"I am very fortunate to have come through that. If anything I think it was my experience which got me through which she will get in time.
"That was just another incredible match after Katie's. She had a big battle too. I am speechless a little bit. I am still trying to come down from it.
"She [Bondar] served absolutely incredible the whole match! The important thing is to remember to breathe and know that I still have a dog at home that loves me regardless. I have a boyfriend too but definitely the dog, that's for sure!
She was playing well so I had to remember that and to not judge myself too harshly and try to break her rhythm as much as possible.
"I am just happy I followed on from Katie as she is doing incredibly well. It is another opportunity we have given ourselves and we will go into tomorrow and put it all out there like we have done so far."
FIRST SINGLES RUBBER REPORT
Katie Boulter gave Great Britain the perfect start to their crucial clash with Hungary as she beat Dalma Galfi 6-4, 6-7(5), 7-6(1) at the University of Bath on Friday.
Boulter, who has led Britain out well throughout the event with victories in both of her singles rubbers, raced out of the blocks, breaking her opponent twice early.
Galfi broke back in the eighth game of the opening set but Boulter won an epic service game to hold and take the lead.
The following set required a tie breaker, with Galfi leading 3-1 at one stage before eventually falling to 4-3 behind. However she kept her cool and found the crucial break to level up at one set each.
As the match wore on to the third set the games became longer and Boulter increasingly relied on the crowd to help keep her focus and give her an extra edge.
Although Galfi forced a tie break she had no match for Boulter as the Brit obliterated her opponent, breaking three times and sealing the victory 7-1.
"Oh my god. This is everything to me." Boulter said on court afterwards. "This means so much. [To the crowd] Thank you so much for helping me out.
I was not calm at all. I wanted to see the last ball go out so I knew I had won. My heart was pumping so quickly.
"My mum is everything to me. My mum, my grandparents, they mean so much to me. Thank you to everyone here supporting me.
"Jo [Johanna Konta] will go out and do the same thing. I hope everyone can be as loud for her as they have been for me."