The shootout which Michael Walther's men lost 4-3 followed a 1-1 draw in normal time.
'Side took the lead on 36 minutes. Roddy Lemba burst through forcing a fine save from Connor Wheatley. However the rebound fell straight to the forward who picked out Ryan Imbert to tap the ball home.
It was Imbert's second goal in two games following Saturday's goal away at Burnham Ramblers.
Experienced keeper Ryan Karrer made a fine save just before half-time to preserve Side's lead, doing brilliantly to keep out a Marcus Bowers free-kick.
Basildon equalised on 64 minutes wuth dangerman Bowers scoring, slotting home after a fine cross by Ashley Blackburn.
The game went to penalties after normal-time as neither team, despite their efforts, could find a winner.
'Side scored their first three spot-kicks through Michael Ogboin, Limbs and Kofi Billzy.
They were leading 3-2 in the shootout as Karrer kept out Jamie Selby-Knight's effort aside of successful penalties by Bowers and Oli Spooner.
Basildon keeper Wheatley then saved Mattheus Santos' penalty before Jack Renew levelled the scores.
James Briggs then saw his attempt saved by Wheatley before George Whitby scored to put Basildon through to the next round.
They now face the holders Takeley at home in the quater-finals.
'Side: Ryan Karrer, Kofi Billzy, Ricardo Bailey, David Etcheri, Joseph Huston-Mendy, Michael Bottone, Roddy Lemba, Jayden Mitchell-Brown, Ryan Imbert (Mattheus Santos), James Briggs, Ben Clarke (Michael Ogboin)
Subs Not Used: Abner Fernandez, Michael Sherwood