With a place in the World Group II play-off on the line, Konta looked to be down and out after losing the second set, having spent almost 10 hours on court over the last four days, and a deciding rubber seemed certain.
She needed to steady herself on her chair before falling to her hands and knees on her way off court, where she was helped by volunteers and paramedics.
But the British number one returned to court seven minutes later after a medical assessment and produced a spirited comeback to earn a 7-6 (1) 3-6 6-2 win, which sealed a 2-0 success in the best-of-three tie.
At the end of the match she was in tears and was joined on court by her parents, who comforted her.
Katie Boulter had earlier put Britain in the lead with her fourth win from four this week, beating Ivana Torovic 6-4 6-4, and Konta eventually completed the job in the Europe/Africa Zone Group 1 play-off.
The fact the former Wimbledon semi-finalist was able to finish the match, let alone win it, says a lot about her spirit and fitness as her time on court totalled nine hours and 54 minutes.
She had back-to-back near-three-hour matches on Thursday and Friday and said that she had an out-of-body experience.
"I started progressively feeling more and more unwell," Konta said. "Light-headed, feeling shaky, feeling a little bit out of body.
"It got the better of me at the end of the second set. I tried not to panic and tried again to assess what I could and do the best I could.
"I was going to go as far as I could."
Krunic must have thought her luck was in when she saw Konta leave the court, having just reeled off five out of six games to level the match.
The Serb was not happy with what happened next, though, and complained of foul play regarding the length of Konta's break.
"It was a very long time," she said. "I don't think it's a fair play, no. But if she had health problems then I can't complain.
"If she almost fainted and she was lying on the ground then it took as long as she needed to get up and play again.
"Regarding her play, I didn't see anything wrong with her on the court. I'm not the one to judge (if she was over-reacting)."
It was Britain's first Fed Cup tie at home since 1993 and it could not have gone much better at the University of Bath.
Boulter says the achievement of racking up four successive victories in front of a home crowd is a career highlight.
"It's definitely up there," she said. "Considering it was my debut as well, it's going to be one I'm going to remember, that's for sure.
"It makes it so special. I came into this week not knowing what to expect and to start my debut and come out with four wins is probably better than I ever imagined.
"I'm doing the best I can every single week of the year, but playing for your country is a little different than just playing for yourself.
"I'm trying to do people proud. It always has a toll on your body no matter what, I feel I've done really well this week with the recovery. I'm feeling as good as I can right now."
The draw for April's tie will be on Tuesday, where GB are again hoping for a home tie.
