Murray, who turned out for his country for the first time in two years, capped a fine weekend with a 6-4 6-2 6-4 victory to give Britain a 3-1 lead and put them in next week's draw for the World Group after a five-year absence.
The Wimbledon champion, who beat 16-year-old Borna Coric on Friday and then teamed up with Colin Fleming to win Saturday's doubles, was far too strong for a weary Dodig who hung on grimly at the end, saving two match points before succumbing.
"I'm looking forward to playing against some of the biggest countries in the world it will be exciting and hopefully I can be part of it," Murray said.
Spain, inspired by the return of Rafael Nadal, needed only two days to crush Ukraine 3-0 to retain their place in the World Group, having been required to face a playoff after losing in the first round to Canada.
Switzerland and the Netherlands beat Ecuador and Austria respectively in their playoffs.
Later on Sunday in the semi-finals, 2010 champions Serbia must try to overturn a 2-1 deficit at home to Canada.
The winners will face the Czech Republic who cruised past Argentina in Prague with a day to spare.


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