The play-off was tied at 1-1 overnight at London's Copper Box after Boulter was unable to take three match points in a heartbreaking defeat to Kazakh number one Yulia Putintseva on Saturday evening.
— Fed Cup (@FedCup) April 21, 2019
The visitors looked set to take a 2-1 lead when Putintseva led Konta by 4-1 and two breaks of serve in the deciding set of Sunday's first rubber, only for the British number one to win the last four games in a 4-6 6-2 7-5 victory.
It was Konta's 11th singles success in a row in the competition and meant victory for Boulter against Zarina Diyas would at last get Britain over a hurdle they have fallen at in four of the last seven years.
It had been in doubt whether the 22-year-old would even take to the court after she picked up an injury during Saturday's match, which a telltale hot water bottle revealed to be to her right hip and lower back area.
She was still using the hot water bottle during changes of ends but, with Heather Watson so out of form, captain Anne Keothavong clearly felt Boulter remained her best option.
That looked a questionable decision when she trailed by a set and a break to 107th-ranked Diyas but Boulter has proved herself quite the competitor in her fledgling career and she battled back to win 6-7 (1) 6-4 6-1.
Diyas had only lost one of 15 Fed Cup singles clashes prior to this weekend but she helped her opponent with a double fault on set point in the second set and Boulter seized the initiative in the decider, clinching victory on her third match point with an ace.
The victory currently secures Britain a place in World Group II, although the format of the competition for next season has not yet been decided.
Proposals for a 12-team finals in the mould of the revamped Davis Cup are reportedly at an advanced stage.