Lawrence rounded his man before lashing home from 25 yards midway through the second half.
Wales laboured to create chances but were rarely threatened themselves in this crucial Group D encounter.
Georgia’s right-back Otar Kakabadze wasted the hosts’ best chance in the last minute of the game when he opted to cross instead of shoot when in front of goal.
The win means Wales can secure second place in the group if Ireland lose against Moldova this evening (7.45m).
However, they may still struggle to reach the playoff spots for Russia 2018 qualifying, with Chris Coleman no doubt keeping an eye on Bosnia’s match with Estonia tomorrow.
Victory keeps Wales second in their group and a point behind leaders Serbia, who play at Austria today before a home tie against Georgia in their final qualification game.
If, as expected, Serbia top the group then Wales will need to be one of the eight best second-place finishers from the nine European qualifying groups for the World Cup.
This win takes them above Bosnia in that mini league table but Wales could still miss out if other results don’t go their way.
Full report to follow…