In tough conditions at Cricklefield Sports Ground the two sides couldn't be separated in the first half with both sides hitting the woodwork and Barkingside's Tom Seldon making a number of impressive saves.
The deadlock though was eventually broken nine minutes into the second period when Hugo Skepelhorn rifled the ball beyond Seldon after Bagasan Graham had missed a glorious one-on-one chance just moments earlier.
But the lead lasted less than six minutes as former Claret, Tyrone Scarlett, got on the end of a free kick from the far right to flick the ball passed Christian Dibble to level matters.
With the game going straight to penalties if the scores remained level City had a number of good opportunities to nick the win in the normal time with Haines hitting the post from a corner, débutante Joe O'Hare blazing over, and both Mark Hughes and Matt Whichelow forcing good saves by Seldon.
Dibble was also called into action towards the conclusion having to make a brilliant fingertip save from Michael Spencer but the two sides couldn't be separated so the tie went to spot-kicks.
Having had penalty woes sporadically throughout the season already City overcame those issues from 12-yards and confidently converted all of their spot-kicks with Hughes, Lee Sawyer, Ben Marlow and Whichelow all converting their penalties.
The hosts saw Scarlett, Spencer and Alfie Wren all find the net with their efforts, but Baff Addae would send his strike wide allowing Luke Daley to step up and send City into the 4th Round.
1. Tom Seldon
2. Jonathan Adebayo
3. Paul Wright (C)
4. Andreas Nikolaoy
5. Alfie Wren
6. Michael Spencer
7. Jonathon Nzengo
8. Baff Addae
9. Luke With
10. George Lataaya
11. Tyrone Scarlett
12. Darren Gray
14. Maliki Toussaint
15. Kemarr Salmon
16. Danny Hayes
17. Darren Behcet
1. Christian Dibble
2. Mark Haines
3. Bagasan Graham
4. Mark Hughes (C)
5. Marvin Ekpiteta
6. Rory McAuley
7. Ben Marlow
8. Harry Morgan
9. Michael Thalassitis
10. Luke Daley
11. Hugo Skepelhorn
12. Matt Whichelow (On for Hugo Skepelhorn 87')
14. Aiden Palmer
15. Lee Sawyer (On for Harry Morgan 90')
16. Damien Green
18. Joe O'Hare (On for Michael Thalassitis 46')
Barkingside: Tyrone Scarlett 60'
Chelmsford City: Hugo Skepelhorn 54'
Barkingside: Baff Addae 40', George Lataaya 87'
Chelmsford City: Mark Haines 83'
Referee: Ray Wells
Assistants: Rhys Battye & Lee Markwick
City Man of the Match: Hugo Skepelhorn