Burns' boot ticked the scoreboard over, and his try before half-time gave Bath a lead they never surrendered.
Will Chudley extended that lead after the interval and Nathan Catt looked to have settled the game, only for his score to be ruled out by the TMO.
Wasps gave themselves hope through Thomas Young's try but too many errors hindered their attempted rescue act.
It leaves Wasps with only one victory from their past 12 matches, but they hold on to fourth place before facing Harlequins at Twickenham on Saturday.
