Needing a victory, Warwickshire started the day fielding and Kent on 37-3.
The visitors took the score to fifty when Chris Woakes got his fourth Wicket when he had Sam Northeast caught by Ant Botha and then in the his ball, Woakes got his fifth when he had van Jaarsveld caught by Laurie Evans.
Tredwell was removed by Carter and with the score now on 54, Woakes claimed his sixth wicket when he had Jones trapped lbw.
Carter took the wickets of Stevens and Mahmood as before the third ball of Rikki Clarke's first over saw Blake who had made 33 bowled and Kent were all out for 111.
Warwickshire struggled when they batted. Varun Chopra was first out after making nine with Darren Maddy soon folllowing without scoring. Ian Westwood went for seven and when Rikki Clarke was out first ball, Warwicks were 17-4.
Troughton added 9, Evans 3 with Johnson out for ten and Botha seven, Warwicks were now 54-8.
Enter Chris Woakes and Neil Carter. The pair added 72 before Carter went for 26 and fourteen runs later, Woakes was out for 51 from 41 balls including nine fours and a six.
Warwicks were all out for 140 and this left Kent 324 to win.
An excellent day for Woakes continued as he took the wickets of Denly (4) and Key (7) as Kent were 13-2 in the fifth over.
Neil Carter took the wickets of Tredwell and Northeast before Woakes picked up his ninth wicket of the match, trapping Jones who had made 41 lbw.
Kent closed on 131-5, still needing 193 runs to win.