Leon Smith's side set up the opportunity to return to the top level of the competition for the first time since 2008 with a brilliant 3-2 victory over Russia in Coventry last weekend.
Wednesday's draw ends a run of seven consecutive ties at home - this will be captain Smith's first away tie - but, with Andy Murray having pledged to make himself available, it is one they will be confident of winning.
Murray has not played in the competition since a victory over Hungary in September 2011 that saw Britain clinch promotion back to Europe/Africa Zone Group I.
In his absence, Dan Evans in particular has stepped up, winning both his singles rubbers to guide Britain to victory over Slovakia last year and then beating top-80 player Evgeny Donskoy in the deciding rubber against Russia on Sunday.
James Ward had previously beaten Dmitry Tursunov to keep the tie alive after Britain had been 2-0 down, and who will get the nod alongside Murray will be a difficult decision for Smith.
Croatia's singles players are likely to be world number 11 Marin Cilic and number 54 Ivan Dodig, who also played the doubles rubber in their most recent match, a defeat by Italy in February.
Murray has won eight of his nine previous matches against Cilic as well as his only meeting with Dodig, while Britain's doubles pair of Colin Fleming and Jonny Marray would also be fancied to win their rubber.
The difficulty for Murray could be that the tie begins on September 13, only four days after the US Open final, where he will hope to successfully defend his title.
Croatia will have the choice of venue and surface and it is not inconceivable the tie could be played on clay, potentially giving Murray little time to adjust to his least favourite surface.
Coincidentally, Britain's only previous tie against Croatia, in September 2007, was also their most recent victory in the World Group play-offs.
On that occasion, a British team featuring Tim Henman in his final Davis Cup tie won 4-1 at Wimbledon, with Murray defeating Cilic in five sets.
Britain could not build on the win, though, losing their first-round match in the World Group against Argentina and then falling to Austria in the play-offs.
The losing streak eventually stretched to five and by the time they won again, in Smith's first match in charge against Turkey in July 2010, Britain had lost to Lithuania and were on the brink of relegation to the bottom tier of the competition.
Thorough preparation and a good team spirit have been the hallmarks of Smith's reign as he has engineered a route back, largely without Murray.
Britain have won the Davis Cup nine times but the most recent of those came back in 1936. Even with Henman and Greg Rusedski, the team did not manage to win a tie beyond the play-offs, and their last victory in the World Group proper came against Spain in 1986.
Britain avoided the likes of Spain and Switzerland, who will take on Ukraine and Ecuador respectively, while other ties include Germany against Brazil and Poland versus Australia.
Smith said on www.lta.org.uk: "Obviously Croatia is a tough draw and especially as it is an away tie.
"In Marin Cilic they have an extremely strong number one player who is currently ranked 11 in the world. Ivan Dodig is their current number two singles player and ranked at 54. But, as we saw last weekend, anything is possible in Davis Cup."