Ireland's record scorer Keane, who recently returned to Tottenham Hotspur after an unsuccessful six-month spell at Liverpool, struck twice in five minutes to lift Giovanni Trapattoni's team level with Italy on 10 points at the top of Group 8.
The home side fell behind for the first time in a competitive game under the Italian when right back Stephen Kelly, in for the injured Paul McShane, failed to deal with a flicked-on Ucha Lobjanidze cross that Karlsruhe striker Alexander Iashvili poked home.
Ireland thought they'd drawn level on 25 minutes after midfielder Keith Andrews's effort from outside the box deflected in off AC Milan defender Kakha Kaladze but an offside Kevin Doyle was harshly adjudged to have interfered with play.
Iashvili had a second goal ruled out just before the hour when he followed up on a Levan Kobiashvili shot that deflected off Shay Given's right-hand post from an offside position.
Ireland were extremely fortunate to equalise from a penalty 17 minutes from time when referee Jouni Hyytia pointed to the spot despite the ball striking Lobjanidze's shoulder.
Keane notched up his 37th international goal five minutes later, heading home an Aiden McGeady corner to put Ireland behind Italy only on goal difference.
The two leading sides meet for the first time on April 1, four days after Ireland host third-placed Bulgaria.
Hector Cuper's Georgia, without a qualifying win in five matches, remain second from bottom on two points.
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