Serbia will face hosts Great Britain in Saturday's promotion play-off after British No.1 Johanna Konta came through a three-set thriller against Hungary's Anna Bondar.
Both Serbia and Great Britain have put together a 3-0 undefeated streak in Bath this week, topping their pools at the eight-nation Europe/Africa Group I event.
Serbia's Olga Danilovic saved a match point to win her singles against Croatia's Jana Fett in a tiebreak 2-6, 6-2, 7-6(7), and then paired up with Aleksandra Krunic to win their doubles rubber against Darija Jurak and Ana Konjuh 1-6, 6-1, 6-4.
Donna Vekic had earlier levelled up the tie when she beat Krunic 1-6, 7-5, 6-1.
Later on Friday, Great Britain ensured they topped Group A after Konta took her first match point in a third-set tiebreak to beat Bondar.
Katie Boulter had earlier needed three sets to beat Dalma Galfi 6-4, 6-7(5), 7-6 (1). The Hungarian, ranked No.311 in the world, had not dropped a set at all during her two previous singles encounters this week.