Bowditch struck fourteen seconds into extra time to seal progress for Ipswich in the first professional meeting of these two clubs.
Although the First Division side dominated for long spells, they missed a number of chances, with striker Darren Bent the leading culprit.
Bent missed a good early chance, robbing Craig Hinton and shooting wide with keeper Stuart Brock well out of goal.
It was not all one-way traffic though with Kidderminster's Graham Ward twice going close with a free kick held at the second attempt by Kelvin Davis and then sending an unmarked header wide.
Jermaine Wright and Bent shot wide for Town before John Williams wasted a great chance for the visitors, rounding Chris Makin but prodding wide with just Davis to beat.
Early in the second half Brock was swiftly off his line to block Bent's shot while at the other end Davis scrambled Dean Bennett's effort round the post.
Martijn Reuser and Bent fired inches wide as Town upped their tempo before match-winner Bowditch joined the fray on 67 minutes.
He soon forced a save from Brock and created a great chance for Bent who smacked an angled shot against the post.
Bowditch did miss a great chance from a Bent pass but he made amends immediately after extra time began, lobbing a loose ball into an empty net when Harriers defender Scott Stamps headed clear unaware Brock was behind him waiting for the ball.
Town continued to dominate the extra period and were only denied a more comfortable passage by the excellent Brock who twice denied both Bowditch and Reuser as well as the unfortunate Bent.
Ipswich: Davis, Makin, Diallo, Santos, Richards, Wright, Miller, Magilton, Reuser, Darren Bent, Counago.
Subs: Price, Wilnis, Naylor, Marcus Bent, Bowditch
Kidderminster: Brock, Stamps, Danny Williams, Hinton, Bennett, Flynn, John Williams, Shilton, Willis, Coleman, Ward. Subs: Danby, Smith, Parrish, Ayres, Lewis.
Referee: Phil Crossley (Bromley).
Attendance: 11,118