Goalkeeper Nukri Revishvili pulled off a string of fine saves to keep Croatia at bay before substitute Otar Martsvaladze stole in behind the defence in the 89th minute and laid the ball off to Kobiashvili who scored with his left foot.
After a scrappy opening Slaven Bilic's Croatia took control with Tottenham Hotspur playmaker Luka Modric dominating the midfield.
Revishvili kept out long-range efforts from Mladen Petric and Tomislav Dujmovic either side of half-time but Georgia came out far livelier after the break, repeatedly hitting Croatia on the counter-attack.
The win lifted Georgia into second place, one point behind Croatia. Greece, in third spot one point behind the Georgians, visit Malta later on Saturday.
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