Bressan, who scored in BATE Borisov's shock UEFA Champions League win over Bayern Munich earlier this month, opened the scoring against Georgia inside six minutes and set up the second goal for Stanislav Dragun.
Dragun served notice of what was to come when he threatened in the second minute, shooting wide, but it was only a further five minutes before Bressan gave the home side the lead with a low right-footed finish from six yards.
The BATE playmaker turned provider in the 28th minute when his delivery from the left channel was headed in by Dragun.
Those two goals put Belarus in control after a first half in which Bressan went close twice more, while Georgia's limited threat amounted to little more than David Targamadze's often optimistic free-kicks.
In the second period, Vitaly Rodionov was denied by keeper Nukri Revishvili and then hit the post as Belarus sought to put the result beyond doubt.
Jaba Kankava headed wide from close range in Georgia's clearest sight of goal, while Alexander Kobakhidze and Levan Kenia were off target from 25 yards.
The much-needed win avenged Belarus' opening defeat in Tbilisi, since when they have lost to France and Spain.