The U’s took the lead after five minutes, as Ruby Sealey worked her way through the defence and slotted the ball past Katie O’Hanlon.
Jenna Legg cut inside and shot just over, while Cecily Wellesley-Smith smacked the post from the edge of the box as United looked to double their lead.
Vitalia Budaieva then saw an effort from outside the area pushed over the crossbar just before half-time.
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The hosts equalised from the penalty spot before the hour mark, as Ella Houghton tucked home following a handball.
Ten minutes later though, Zoe Barratt won a free kick just outside the box, and the forward sent a rocket into the goal.
Billericay equalised again in the 80th minute, as Breon Grant cut inside and unleashed a strike which found the net.
The tie went into extra-time, and Paris Sercombe made a fantastic double save to deny a Billericay counter-attack, before the hosts then seemingly won the clash when Houghton netted in the 120th minute.
It was United’s turn to grab a leveller, and did so in the 128th minute when Legg squared the ball for Sarah King to smash home.
Sercombe saved the first penalty in the shoot-out, but then Jaime Gotch hit the crossbar.
Billericay scored their next three penalties, while Nic Gibson also hit the crossbar, allowing the hosts to then make it 4-2 in the shoot-out.