The Bees started in the same shape they ended away to Forest Green on the weekend and began brightly, David Tutonda striking the upright following an effort from the edge of the box.
Chances followed for Coulthirst, Akpa Akpro and Elliott Johnson before Barnet took a well-deserved lead in the 22nd minute.
Mauro Vilhete’s corner was headed down by Santos at the far post to Jean-Louis Akpa Akpro who fired high into the top corner, sending the 155 travelling Bees fans jubilant.
Peterborough started to find their way back into the game, Tafazolli and Marriott both having chances before they equalised on the half hour mark.
Gwion Edwards cut inside and shot at goal, beating Jamie Stephens at his near post to make it 1-1.
Shortly afterwards, Stephens pulled off a brilliant save to deny Forrester who was racing down on the Barnet shot-stopper.
In the 36th minute, Coulthirst sent through Jean-Louis Akpa Akpro who had Taylor racing up with him. The Frenchman opted to go it alone and his shot was saved.
Just 3 minutes later, Barnet regained the lead. The Bees opened up the Peterborough defence with some tricky, one touch passing. Akpa Akpro laid it off to Mauro Vilhete to slot home.
The second period saw a resurgence from Peterborough, who had opportunities from Marriott, Maddison and Tafazolli all dealt with by the Barnet back line in the first 10 minutes of the half.
56 minutes in and Barnet were awarded a free kick on the edge of the area after Coulthirst took a boot to the face. Vilhete stepped up and it struck the arm of a man in the wall and the referee awarded a penalty to the Bees.
Coulthirst stepped up only to see his spot kick saved by Bond, but the violent backspin sent it across goal and in, making it 3-1 to Barnet.
It was now a matter of holding out for the Bees, with Peterborough throwing everything at them. The defence stood firm, with Jamie Stephens pulling off a number of wonderful saves from close range to deny any chance for the hosts.
The Bees will now patiently wait for Thursday’s Second Round Draw.
Barnet: Stephens, Johnson, Santos, Nelson, Campbell-Ryce (Bover 55′), Coulthirst (Amaluzor 68′), Tutonda, H. Taylor, Fonguck (Clough 88′), Vilhete, Akpa Akpro.
Unused subs: Ross, Akinola, Shomotun, Mason-Clark