There was an enthusiastic, season-high crowd of 257 to cheer the Shrimpers on and their team came out determined to upset the odds.
Bowers were fielding an almost full-strength team for the match and the game started with Harwich on the defensive but not allowing the visitors to get a clear sight of goal.
Harwich settled down to come back and take the initiative. They deservedly took the lead in the 30th minute after Liam Riley's free kick was not cleared and the alert Dan Keeble followed up to net from close range.
Riley was a threat with his well-placed corners and from one of them Joe Knight headed narrowly wide of the target. There was further excitement for the home fans when Jack Wood's header was spilled but Bowers keeper Rhys Byrne recovered well to save.
Bowers had a big chance just before half time when Stan Skipper's long throw saw Samir Ali create an opening but Alex Temori's effort went wide.

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Bowers started the second half with a 15-minute spell in which they impressed by putting the Harwich defence under pressure and were rewarded with an equalising goal with Temori's looping shot being diverted past goalkeeper Brad Cook.
Harwich manager Matt Carmichael's response was to bring on 3 substitutes to freshen up his team.
The Shrimpers had matched their opponents throughout and with 15 minutes remaining they retook the lead. Again it came from Riley's precision set-piece - a corner this time - that was headed across the goalmouth by Keeble for Ewan Livingston to net from close range.
Bowers threw caution to the winds in the closing stages with goalkeeper Byrne joining the attack. In the closing seconds, Cook pulled off a fine, full-length diving save before the final whistle produced joyful scenes as the Harwich players joined their fans to celebrate rolling back the years in their historic club's cup fighting traditions.
It was their first win over a side three steps higher in the pyramid in 34 years, the last such occasion coming in 1990 when Fisher Athletic were beaten 2-1 on the Royal Oak in the FA Cup.
The draw for the last 16 takes place on Friday.