Wales had not won since thrashing Italy in the Six Nations on February 1, but a youthful side ran out 18-0 victors in the second round of fixtures in the Autumn Nations Cup.
Rees-Zammit also set up a try for replacement scrumhalf Rhys Webb and flyhalf Callum Sheedy scored his first points in a Wales jersey with a conversion and two penalties as the home side laboured to a victory that will ease the pressure on coach Wayne Pivac.
Wales remained third in Group A, five points behind leaders England, who they host next Saturday, and they handed debuts to flanker James Botham, grandson of former England cricketer Ian, scrumhalf Kieran Hardy, centre Johnny Williams and teenage flyhalf Ioan Lloyd.
Georgia showed improvement from their 40-0 drubbing by England last time out, but were surprisingly outmuscled in the scrum and have now scored only seven points in their last three tests.
Additional reporting by Reuters.