Kent resumed on 121-5 and it was Neil Carter and Chris Woakes who went on to record five wicket hauls.
Carter had taken one of the five wickets to fall in Kent's second innings yesterday afternoon and Carter took two wickets in four balls to remove Stephens and Blake.
Kent were now on 152-7, needing 324 to win and although Martin van Jaarsveld was the only form of resistance with a superb 110 from 160 balls including 16 fours, Woakes took the wickets of Mahmood and Cook in the space of three balls to give him eleven wickets in the match.
Carter had Coles caught to give him a tally of eight wickets in the match and Kent were all out for 228 and a much needed win for Warwickshire by 95 runs.