All the action happened in a five-minute spell at the start of the second half, Kuram Gashia putting the visitors ahead in the 57th minute before Finland lost Alexei Eremenko to a red card two minutes later.
Hamalainen nevertheless popped up to equalise for the hosts in the 62nd minute as Finland picked up their first point in Group I.
There was little to choose between the teams in the first half, Tim Sparv failing to sent his header goalwards from six yards for the hosts.
Georgia almost went ahead five minutes before the break when David Targamadze's corner was met by Kashia in the six-yard box and the defender's header was tipped round the post by Niki Maenpaa.
Sparv was again involved after half-time, his header punched clear by Nukri Revishvili but it was the visitors who went ahead, Kashia slotting home Targamadze's corner 12 minutes into the second period.
Two minutes later Finland's task intensified when Eremenko was sent off after his second booking for handball.
However, Finland levelled in the 62nd minute when Hamalainen slotted home from close range .