Heather Watson and Johanna Konta both posted straight-sets singles wins to give GB an unassailable 2-0 lead over minnows Liechtenstein ahead of the doubles rubber.
Konta kicked things off with a 6-0 6-0 thrashing of Kathinka von Deichmann before Watson took to the court to see off Liechtenstein???s number one player Stephanie Vogt 6-2 6-3.
Judy Murray???s team are competing in Europe/Africa Zone I ??? the third tier of the global competition ??? and need to win their four-team group in order to keep their hopes of promotion alive.
They will next face Turkey on Thursday before what is likely to prove a decisive clash with Ukraine on Friday.
Ukraine did have star player Elina Svitolina in their line-up for Wednesday???s clash with the Turks. She was a late arrival in their camp.
However, former world number one Victoria Azarenka did not play for Belarus. She made it to the Hungarian capital but her equipment, including racquets, did not so she sat out her side???s meeting with Georgia.
Belarus and Great Britain will meet in Saturday???s play-offs if both teams win their group.