Despite any clear chances, the home side constantly threatened in the opening period with Giorgi Kvilitaia causing Ireland problems on the counter attack.
Ireland ended the opening 45-minutes strongly with chances in the final minutes, but it remained scoreless at the break.
Both sides continued to press after the break, but neither Darren Randolph or Giorgi Loria were overly tested.
In the 78th-minute, Aaron Connolly came on for his senior international debut, and had an opportunity to score in the dying moments but was unable to convert.
Ireland now travel to Switzerland to play the group’s top seeds in Geneva on Tuesday.
Georgia: Giorgi Loria; Otar Kakabadze, Guran Kashia, Gia Grigalava, Jemal Tabidze; Jaba Kankava (capt), Otar Kiteishvili (Giorgi Aburjania 90); Tornike Okriashvilli (Elguja Lobjanidze 79), Jano Ananidze, Valeri Qazaishvili; Giorgi Kvilitaia (Levan Shengelia 73)
Republic of Ireland: Darren Randolph; Seamus Coleman (capt), Shane Duffy, John Egan, Matt Doherty; Jeff Hendrick, Glenn Whelan, Conor Hourihane (Derrick Williams 90+3); Callum Robinson (Alan Browne 73), James Collins (Aaron Connolly 78), James McClean
Referee: Marco Guida (ITA)