After failing to find a cutting edge in regulation time, both teams offered a greater attacking threat in the additional 30 minutes.
Konstantinos Mavropanos rattled the woodwork with a powerful header, before Zuriko Davitashvili was denied by Odysseas Vlachodimos from point-blank range, ensuring penalties were required to decide the winner of the qualifying playoff final.
Georges Mikautadze's penalty miss did not prove costly as Anastasios Bakasetas and Giorgos Giakoumakis both failed to convert from the spot to send Georgia to their first-ever major tournament.
Georgia and Greece were participating in the playoff final after securing victories over Luxembourg and Kazakhstan respectively.
With so much riding on the contest, a fear of losing resulted in a cagey half where the respective goalkeepers had very little to do.
In fact, the first meaningful shot did not arrive until the closing stages of the first period, with Georgia's Giorgi Chakvetadze forcing Vlachodimos to tip over from a whipped free-kick.
While clear-cut chances may have been at a premium, tensions still managed to boil over following the conclusion of the first half.
A melee between both sets of players ultimately resulted in a red card being shown to Georgia's unused goalkeeper Giorgi Loria.
The half-time interval did little to change the pattern of the nervy encounter, as two strong defences continued to nullify any attacking threat.
The lack of goalmouth action was demonstrated by the fact there was just one shot on target before referee Szymon Marciniak called an end to regulation time.
Greece moved through the gears in the first half of extra time, with Tasos Bakasetas's fierce shot drawing a superb diving save from Giorgi Mamardashvili.
Konstantinos Mavropanos then went agonisingly close to breaking the deadlock, rising high to meet a corner and send his thumping header crashing against the crossbar.
Those near-misses persuaded Georgia boss Willy Sagnol to call for reinforcements, introducing Zuriko Davitashvili and Nika Kvekveskiri.
Within seconds of his arrival, Davitashvili threatened to find the all-important breakthrough, only to see his first-time strike kept out by Vlachodimos's instinctive reactions.
Following a short half-time break, Lasha Dvali produced a purposeful driving run to force the Greek goalkeeper into action at the near post.
With neither team able to find a winner, a penalty shootout was required to decide who would book their ticket to Germany.
Bakasetas saw his spot-kick saved before Georgia's Mikautadze sent his penalty harmlessly wide of the goal to give Greece renewed hope.
However, Giakoumakis was unable to direct his effort on target, allowing Nikoloz Kvekveskiri to convert the winning penalty and spark wild scenes of jubilation in Tbilisi.
While Greece's 10-year wait for a major tournament appearance will go on, Georgia can look forward to participating in a Euro 2024 group alongside Portugal, Turkey and the Czech Republic.