Johanna Konta and Heather Watson both won as Britain beat Hungary to reach the Fed Cup World Group II play-off.
World number 11 Konta broke Fanny Stollar's serve five times in a 6-3 6-1 win to give GB an unassailable 2-0 lead and no need to play the doubles rubber.
Earlier, Watson came from a set down to beat Dalma Galfi 3-6 6-1 6-4.
The British number two was broken in the eighth game of the deciding set after wasting two match points, but held her next service game to win.
After topping their group in the Europe/Africa Zone, Konta and Watson won their third successive matches in Estonia to put captain Anne Keothavong's team into April's World Group II play-offs for the second straight year.
"It was a brilliant team effort," said Keothavong. "On paper in all the matches they played this week they were favourites but they delivered in style.
"I hope this is the year we get a home tie. It's been a great effort to get out of this zone and I don't want to be back here next year."
It will be the fourth time in the past seven years that the British team have been one step away from reaching the elite World Group level.
They lost last year's tie to Romania, when Ilie Nastase's shocking behaviour overshadowed a 3-2 win for the hosts, while the British team also lost in Sweden in 2012 and Argentina in 2013.
GB will find out who they play in April when the draw is made on Tuesday, 13 February at 10:00 GMT. The rankings, which will be updated on Monday, will determine whether they are seeded or not.
The draw will also decide which team plays at home - something Britain have not done since 1993.