Goals from Ryan France, Jamie Forrester and Ryan Williams deservedly sealed City's passage into the next round of the competition with defender David McGurk claiming a late consolation for the hosts.
In an entertaining first half both sides had plenty of opportunities to score with Quakers winger Alan Morgan going close.
Moments later the home side were let off the hook when Michael Price made a hash of France's cross, under pressure from Forrester but thankfully for the Darlington keeper Craig Liddle was alert to clear the danger.
City eventually broke the deadlock in the 36th minute when Williams swung over an inviting cross that France headed home at the back post.
Forrester made it two with half time approaching receiving a long ball from Steve Burton before lifting the ball over the onrushing Price.
The home side were given some hope with ten minutes of the game remaining when Mark Sheeran pulled a goal back prodding the ball home after a goalmouth scramble.
However, any hopes Darlington had of finding a way back into the game were dashed a minute after Sheeran's strike as Williams cut inside the box before placing the ball past Price with a confident finish.