Shane Duffy took advantage of an error by keeper Giorgi Makaridze to head the visitors in front in the fourth minute.
But the Republic sat back and the hosts equalised before the break through a well-worked goal by Valeri Qazaishvili.
Martin O’Neill’s side stay second in Group D but are now two points behind leaders Serbia, who beat Moldova 3-0.
James McClean and Aiden McGeady missed chances late on but victory would have flattered a lacklustre Irish side, who face Serbia at home on Tuesday.
Wales closed the gap on the Republic to two points by beating Austria 1-0 later on Saturday.