The 19-year-old forced a 90th-minute save from keeper Giorgi Loria and then fired into the side-netting when he might have done better in a lightning cameo which almost paid rich dividends.
Defender Shane Duffy, who had led a concerted rearguard action for much of a tight Euro 2020 qualifier in Tbilisi, headed straight at Loria from an injury-time corner, but Ireland ultimately had to make do with a 0-0 Group D draw.
In truth, they deserved no more from a performance which, barring a late onslaught, lacked inspiration against a Georgia side which was far more creative, but could not find the cutting edge to reap the rewards of their good work.
The Republic's qualification hopes were dealt a blow as a result of the two priceless points left behind at the Boris Paichadze Arena with Tuesday's already difficult trip to Switzerland now assuming even greater proportions.
Having endured desperately close encounters on their last two trips to Tbilisi, Ireland knew exactly what to expect, and they found themselves back-pedalling within seconds of the kick-off when Valeri Qazaishvili exchanged passes with Jano Ananidze on the edge of the box before screwing a shot wide.
However, Georgia boss Vladimir Weiss' worst fears were almost realised with just four minutes gone, if from an unexpected source.
The Slovakian had admitted his concern at Duffy's potency from set-pieces, but it was John Egan who got to Conor Hourihane's free-kick to glance a header across the helpless Loria, only to see the ball come back off the post.
Georgia's assault continued apace with Otar Kiteishvili orchestrating from the middle of the pitch behind the lively Ananidze and Qazaishvili, prompting a determined rearguard action from the visitors, who were creating little of note in response.
Strong in the tackle and decisive on the ball, the home side pressed Ireland back into their own half to leave striker James Collins painfully isolated in his first start for his country.
Matt Doherty, playing at left-back in the absence of the suspended Enda Stevens, threatened to change the flow of the game when he embarked upon his first run of note with 27 minutes gone, surging past Georgian defenders on his way into the penalty area, but he was unable to deliver his cross before the ball ran out of play.
The traffic remained largely one-way and the Republic were indebted to Duffy, whose presence in Tbilisi was in doubt until Wednesday because of a calf injury, and he repelled a series of crosses from full-backs Jemal Tabidze and Otar Kakabadze.
But for all their possession and intent, Georgia were unable to create clear-cut opportunities with Qazaishvili dragging a 42nd-minute effort from distance well wide at the end of a counter-attack launched by striker Giorgi Kvilitaia after James' McClean had underhit a pass to Glenn Whelan.
Mick McCarthy's men continued to defend with their traditional stubbornness after the break with Egan matching Duffy's aerial prowess inside his own box, although keeper Darren Randolph was relieved to see Ananidze's 66th-minute free-kick sail high over his crossbar.
McCarthy made his move with 11 minutes remaining when he sent on Connolly for his first senior cap, and the youngster might have won it at the death with a low drive which prompted a solid save by Loria, before blasting into the side-netting and then Duffy's big chance.