The Scot was broken in his first service game and Fognini's aggressive approach meant Murray was constantly under pressure.
But errors began to creep into Fognini's game and the more desperate he became to produce winners, the more mistakes he made.
Murray was not at his best either but his steady play soon brought him level and left the temperamental Fognini close to boiling point.
The Italian twice hurled his racquet to the ground after playing wayward shots while 5-4 down and Murray wrapped up his win when Fognini hit a limp smash into the net when facing his second match-point.
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