After securing back-to-back wins over Italy on Saturday for a 2-1 lead, Britain required one victory to reach their first Davis Cup semi-final since 1981, but Fognini was in inspired form on the Naples clay as he cruised past Murray 6-3 6-3 6-4.
With the scores level at 2-2, it was left to the world number 161 Ward to claim an unlikely victory, but he went down in straight sets to Seppi 6-4 6-3 6-4.
World number 13 Fognini, who has won all three of his career ATP singles titles on clay in the last nine months, punished Murray's sloppy serve throughout the match and he recorded a routine victory over the Wimbledon champion.
Seppi, world number 34, was rarely troubled in the decisive fifth rubber by his lowly-ranked opponent and, after taking the first two sets in simple fashion, he comfortably saw out the third to give Italy a 3-2 victory.
Italy will play Switzerland or Kazakhstan in September's semi-finals, their first appearance at that stage since 1998.
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