Barnet had taken the lead through John O'Flynn after 11 minutes when he was teed up by Michael Hughes. However, within four minutes Charlton were level when Nick Bailey set up Izale McLeod, whose well struck shot took a deflection and flew past Barnet keeper Jake Cole.
The visitors continued to give as good as they got, and saw a penalty shout waved away moments after McLeod's leveller, but Charlton were not to be denied and it was Tuna who struck the decisive blow.
After being set up by a perfect pass from Therry Racon, he raced clear and beat Cole with a fine finish into the bottom right-hand corner.
Charlton took control as the half wore on and effectively sealed their place in the regional last eight when Bailey netted from the penalty spot, after McLeod had been fouled in the area by substitute Joe Tabiri.
The hosts managed to bag a fourth before full-time and McLeod was again involved, setting up Scott Wagstaff for a charging run and fierce shot which went in off the underside of the crossbar.