It was full back Richards first goal for the club in his 78th appearance.
Boss Russ Penn hopes the hard earned victory will kickstart the kind of cup adventure his National League North troops experienced last season when they knocked out visiting Championship side Reading before suffering a heartbreaking 2-1 extra time defeat at home to Premier League West Ham United in round four.
Harriers made one change from the side which drew 0-0 in a National League North clash against AFC Fylde at Aggborough earlier in the week with midfielder Zak Brown coming in for Nathan Lowe who was ruled out with a knee injury.
Academy winger Samson Hewett was named among the substitutes.
Harriers were soon pushing forward and forced the first corner of the game in the seventh minute after a shot from Ethan Freemantle had been blocked by Alex Collard. The flag kick by Brown, however, came to nothing.
Another corner, three minutes later, was this time taken by captain Shane Byrne and also proved unproductive.
Harriers were awarded a free kick 30 yards out in the 22nd minute, but Ashley Hemmings’ shot sailed harmlessly over the bar.
Southern Premier Central side Tamworth were rarely in the picture as an attacking force and shortly after the half hour mark Byrne sent over a corner which was headed over by Nat Knight-Percival.
Home goalkeeper Tom Palmer was finally called into action couple of minutes later when he safely held on to a snap header from Ty Deacon.
Kidderminster fashioned their first serious effort on goal on 37 minutes when Byrne set up the chance for Freemantle to fire in a shot which was comfortably saved by former Aggborough goalkeeper Jasbir Singh.
The second half began with Harriers striving to find a clear-cut way through the stubborn Tamworth rear-guard.
Hemmings tested his luck with an angled shot which flew high over the bar before he tried again shortly afterwards with another effort which this time was closer to the target.
Harriers started to ring the changes shortly before the hour mark with Amari Morgan-Smith and Gabby Rogers replacing Freemantle and Yusifu Ceesay.
Rogers was soon in the thick of the action with a cross which found Joe Foulkes whose header was well tipped over the bar by Singh.
Harriers’ efforts were finally rewarded when Keziah Martin set up Foulkes for a cross which was headed home by Richards in the 61st minute.
Soon after Byrne had a shot well saved by Singh before Rogers and Morgan-Smith went close with fierce drives.
Kidderminster: Palmer; Foulkes, Pearce, Knight-Percival, Richards; Byrne, Brown (Margetson 87); Cessay (Rogers, 57), Hemmings, Martin; Freemantle Morgan-Smit, 57). Subs not used: Emery, Leak, Lissimore, Hewett.
Tamworth: Singh; Curley, Collard, Dolman, Cockerill-Mollett (Cater, 79); Fairlamb, Mussa, Milnes (Hurst, 63), Finn; Clement (Jones, 67); Deacon. Subs not used: Fallows, Hart, Howkins, George.
Referee: Andrew Humphries.
Attendance: 1,663 (201 away fans)