Henman, the British No 1, dropped the second set against Daniel Omaboe, but went on to gain a 6-3, 4-6, 6-0 victory, while Milligan beat Isaac Donkor, 6-4, 6-3.
Following their victory, Britain will now meet either Egypt or the Ivory Coast at home, probably at Eastbourne, in September and the winners of this contest will be promoted to Euro-Africa group one.
If Britain are promoted, winning group one would give them a further promotion to the world group of the top 16 nations.
Monica Seles, playing for the first time for the United States in the Fed Cup, led her country into the final of the event with an overwhelming 6-0, 6-2 win over Kimiko Date, of Japan, in Nagoya yesterday. The United States then wrapped up a 5-0 win when Lindsay Davenport beat Ai Sugiyama, 7-6, 7-5, before Davenport and Linda Wild beat Sugiyama and Kyoko Nagatsuka, 6-2, 6-1, in the doubles rubber.
Seles gave the ninth-ranked Date no chance to repeat her giant-killing performance in the Fed Cup quarter-finals when she beat Steffi Graf, the world No 1. On Saturday, Seles had beaten Ai Sugiyama, 6-2, 6-2, and was even more impressive against Date, who appeared sluggish and had still not fully recovered after reaching the semi-finals at Wimbledon. Date had lost the opening match of the event when she fell to Davenport, 6-2, 6-1, on Saturday.