The home side looked nervy from the start but booked their place in the third round Cup with two goals from Paul Benson and one from Ben Strevens.
Harriers started brightly and came close to opening the scoring in the ninth minute when Justin Richards lost his marker on the edge of the box to fire just wide of the Daggers goal.
Kidderminster's pressure paid off in the 13th minute when Mark Creighton took advantage of slack marking in the Dagenham box to fire in a bullet header from close range.
Dagenham full-back Scott Griffiths tried to level the score from long range in the 23rd minute only for Harriers keeper Scott Bevan to save well.
Midfielder Glen Southam came even closer in the 29th minute when he powered a volley straight at Bevan from 25 yards.
The home side's pressure paid off in the 33rd minute when Benson lost his marker in the edge of the area and fired into the roof of the net.
Kidderminster almost caught Daggers with a sucker punch in first-half stoppage time, when midfielder Russell Penn shot inches wide of Tony Roberts' goal from 20 yards.
The home side came flying out of the blocks in the second half with Southam forcing a fine save out of Bevan with a long-range shot in the 46th minute.
Southam came close again minutes later when he hammered the ball into the side netting from six yards after powering into the Kidderminster box.
Daggers had Roberts to thank for keeping the scores level in the 63rd minute when he pulled off two instinctive saves to deny Iyseden Christie and Richards from close range.
The former Wales international keeper came to the rescue again in the 78th minute when he blocked at Richards' feet when one-on-one with the striker.
The home side weathered the storm and finally broke the stalemate with five minutes left when Strevens drilled a Jon Nurse cross past Bevan at close range.
Benson sealed the win in injury time when he slotted home from six yards after bursting into the Kidderminster box.