Andy Murray beat Sergiy Stakhovsky of Ukraine to preserve Great Britain's Davis Cup Euro-African Zone One status.
Murray's second singles win of the weekend sealed Britain's win
Great Britain led 2-1 overnight, with two singles matches to go.
Murray was not at his best but his class proved the difference as he won 6-3 6-2 7-5 to take an unassailable 3-1 lead in the tie and relegate the hosts.
In the final match Jamie Baker was given his debut against Sergei Bubka, losing the dead rubber 6-3 7-6, but Britain won the tie overall 3-2.
Bubka, son of the legenday pole vaulter, trailed 4-0 in the second-set tie-break after Baker had fought back from a break down.
But Bukba recovered to win the tie-break 8-6 and close out the match.
Winning any Davis Cup match means a lot because it is a great honour to represent your country
Afterwards, Murray admitted it had not been a vintage display, but said he was glad to have helped the team get the result they needed.
"I did not play at my best but it was enough to win," he said.
"Winning any Davis Cup match means a lot because it is a great honour to represent your country and it's the first time I have had to play all three games."
After the match, he was seen clutching an icepack to his chest, but he played down fears this might rule him out of the Bangkok Open.
"Every time I breathed I was feeling a shooting pain in my chest," he said.
"It happened early in the third set but hopefully I will be OK for next week."
Ukrainian number one Stakhovsky, ranked 188 in the world, refused to give in against a lacklustre Murray, chasing down every ball on the court as he tried to respond to the Odessa crowd's cheers.
But, in truth, the result was rarely in doubt as Murray ruthlessly extinguished all but one of Stakhovsky's break points with a succession of winners.
Despite holding his opening two service games to love, Stakhovsky always looked in trouble once two Murray cross-court winners had earned the Brit a crucial first break in game six.
That was enough for the 19-year-old to take the opening set and, having held off a brief Stakhovsky revival early in the second, Murray broke the home favourite twice in succession to take the second set.
Stakhovsky had never won a five-set match in his career to date and, perhaps buoyed by that knowledge, Murray adopted an even more casual approach to the third set.
Once again this gifted Ukraine a couple of break points in game two but, when it mattered most, Murray always found the answer and he underlined his superiority, after a needless disputed line call, by breaking in game five.
Ironically, after two hours and with break points reaching double figures, Stakhovsky dug deep to break Murray's serve to love in the very next game.
However, once again Murray knuckled down when it mattered most, breaking crucially at 5-5 after a ferocious exchange at the net, and he saved two more break points to serve out and seal victory in the match and also the tie.