Warwicks had made 476-9 in their first innings but Rogers led the Derbyshire fightback and at stumps was 237 not out from 323 balls which included one six and 31 boundaries.
Chris Martin and Ian Salisbury took a wicket each in the opening two sessions of a cloudy day. In the final session of the day, Chris Woakes struck to take two wickets in three balls with Smith (27) and two balls later catching and bowling Clarke for a duck.
Only Wicket keeper Tim Ambrose and Ian Westwood did not get a turn at bowling.Westwood took two catches off the bowling of Salisbury (4-95) just before the close which saw Derbyshire move their total on to 462-7.
In the One Day international at The Oval, Ian Bell made 73 in England's 297-5 total. South Africa were skittled out for 170 to give England a win by 126 and a 3-0 series win with two matches to play.