Heather Watson and Johanna Konta won their singles matches against Turkey to take an unassailable lead in their final best-of-three group match.
Jocelyn Rae and Laura Robson completed a 3-0 sweep in the doubles, with Anne Keothavong's team topping Group C.
Their play-off place was already sealed after Latvia earlier beat Portugal in Britain's group.
Watson powered Great Britain to a 1-0 lead by beating 163rd-ranked Ipek Soylu 6-0 6-1.
British number one Konta then overcame an early wobble to beat world number 86 Cagla Buyukakcay 5-7 6-4 6-3 in the second singles match.
"I didn't get the chance to play Fed Cup last year, but from the very beginning of the season I was clear that I wanted it to be part of my schedule," Konta told British Tennis.
"I think whenever I get an opportunity to represent Great Britain in a team environment, I look to take it."
Rae and Robson then breezed to a 6-2 6-2 victory over Ayla Aksu and Pemra Ozgen in 58 minutes to ensure Great Britain ended the group stages unbeaten.
Unlike the men's team competition, the Davis Cup, which has a World Group of 16 nations, the Fed Cup divides its top teams into two groups of eight - World Group I and World Group II.
The 91 nations outside the top tiers are divided into three regional zones and Britain have one chance per year to escape - a format that hugely frustrated former captain Judy Murray.
The Europe/Africa Group I event, which this year takes place in Estonia, has 14 teams divided into groups, with Poland, Croatia, Britain and Serbia the seeded nations.
Four group winners will progress to promotion play-offs on Saturday, and two nations will then qualify for World Group II play-offs in April - which could see Britain given a home Fed Cup tie for the first time since 1993.
They fell at the same stage in 2012 and 2013 - away ties in Sweden and Argentina - under the captaincy of Judy Murray.
Great Britain will face second seeds Croatia in the Fed Cup promotion play-off on Saturday.