Two early goals and then a disciplined performance was enough to ease them through to the next round where they have been drawn at home to play a very good Mossley Hill side.
That game is due to be played on 5 October but before that Billinge boss, Wayne Wardle will reflect on this good win.
The Storks were first out of the blocks with wide-men Ryan Platt and Kyle Ross combining well with centre-forward Rob Lamont.
In fact, Lamont had two good chances in the first five minutes but Liver keeper Allen Weston was equal to his shots.
At the other end Storks' keeper, Kai McAlister saved well from Luke Mason.
On twelve minutes Ross produced a superb run and cross that Weston just about dealt with before Ross then found James Dagnall with a neat pass; the overlapping full-back produced an excellent cross that just evaded the onrushing Lamont and Platt.
On 16 minutes Billinge were ahead when Weston failed to deal with a wicked in-swinging Ross corner.
The Academy custodian could only punch the ball straight onto the head of Zak Riley who forcefully headed home.
The Storks now had the bit between their teeth as Ross went on a mazy run only to be denied by a good defensive block and then a moment later the same player earned the home side a penalty.
The winger cut in from the right and hit a left-footed shot into a crowded area; the ball was stopped by two Liver defenders but both the referee and nearside assistant instantly signalled a penalty.
Billinge's captain Chris Cottington stepped up to hammer the ball home and double the advantage.
Ross and Lamont both went close, and then on the half-hour mark left-back Liam Thomas went on a fantastic run from his own half.
Taking the ball into the area it looked to all the world that he was brought down for a penalty but referee Mark Shaw ignored The Storks' claims.
It looked more of a penalty than the one he had given.
Lamont, Nial Dawson and Danny Murphy all had efforts on goal for Billinge but they just could quite find that third goal.
At the other end, Mason had a great chance but hit his shot high and wide with just McAlister to beat.
The final action of the half saw Riley connect with a Dawson corner but this time the hardworking midfielder hit the bar with his header.
The second-half, in the main, was a niggly and messy affair but it was a credit to Billinge that they kept their heads and defended - as a team - superbly.
McAlister was never tested as Cottington and Aaron Rice were a rock in the centre of defence and cleaned everything up.
Meanwhile at the other end Lamont - who worked hard all afternoon - misdirected a header when he should have done better and was denied at the last moment after a fine run into the box.
On 80 minutes the excellent Dawson showed a great first touch and turn to lose his defenders only to be a denied a goal by an excellent save from Weston.
The final action of the game when Cottington went close with a shot and then in the final minute Thomas produced another tricky run into the six-yard area only to be denied by a fine tackle before he could get his shot away.
After the game, The Storks’ boss Wardle said: "We knew it would be a difficult match and I thought we managed the first half superbly.
"The second period we were not so good but I thought we defended well and we now look forward to the next round."
Billinge are in cup action again next Saturday when they take on Egerton at home in the JB Parker Premier Division Challenge Cup.
Kick-off at Barrows Farm is 3pm.