The striker hit the winner in Saturday's sensational 3-2 comeback against MK Dons and bagged his side's second against the Addicks after Guly Do Prado had opened the scoring.
The visitors to St Mary's should have gone ahead in the eighth minute. Former Saint Bradley Wright-Phillips was inches away from connecting with the impressive Therry Racon's cross, which Scott Wagstaff somehow skied from six yards.
Two minutes later Brazilian winger Do Prado ran onto Barnard's pass to smash the ball past Charlton goalkeeper Rob Elliot.
Determined Saints then struck the decisive blow with less than an hour gone. Do Prado repaid Barnard for his earlier assist, crossing for the well-placed forward to tap home his 13th of the season.
Adkins' men eased to victory and are now two points off second-placed Huddersfield with two games in hand. The only negative for the south-coast side was the loss of new signing Dale Stephens to injury just before the interval.