Basildon took the lead after 15 minutes when Shrimpers goalkeeper Brad Cook was unable to reach a high cross and Kunie Aideymi was well placed at the far post to net.
Harwich lacked cohesion and struggled to get their game together during the first half.
However, they did have two really good chances to score during this period. Both fell to Luke Mayhew, usually an ace marksman for the Shrimpers. His first opportunity was well saved by the Town goalkeeper and when his pace took him into a one-on-one situation with the keeper, this time his shot went the wrong side of the post.
Town deserved their lead but Harwich came out for the second half determined to put things right and equalised within minutes of the restart.
Theo Clayson provided the cross and recent signing Shaun Avis (pictured) was on the spot to net with a good finish.
The second Harwich goal soon followed and was a beauty - an unstoppable rocket shot from Louie Bloom.
Harwich were much better balanced now, with Liam White, back after a lengthy injury absence, impressive in midfield and they were well on top by the halfway stage of this half.
Basildon tried to comeback but Harwich defended well but needed a third goal to make victory secure.
It came in the 87th minute when Mayhew, not on the scoresheet but always dangerous, did well to set up James Letts, who had positioned himself perfectly for the pass and was able to comfortably slot the ball into the home team's net.
The Shrimpers now take their place in the draw for the last 32 which will be held on Friday during the BBC Essex Fans' Phone-In between 6 and 7pm.
All the big guns are now in the hat, including EFL side Colchester United and National Leaguers Southend United. Neighbours Little Oakley are also potential opponents.
Harwich team: Brad Cook, Tom Watkins, Jack Wood, Ted Johnson, Tiwa Odufuye, Tyler Sullivan (capt), Alfie Canning, Luke Mayhew, Louie Bloom, Shaun Avis and Theo Clayson. Substitutes: Liam White, James Letts (both used), Dan Keeble, Oakley McNair and Charlie Canham (not used).