The match started off slowly but fly-half Daniel Kirkpatrick found space in the Dragons defence and went over for the first try of the afternoon.
Palis crossed for a try of his own, before Jones added three quick penalties after Palis converted his own try.
Jones converted both tries but Thomas Combezou crossed for Castres, with Palis converting and then adding a further penalty to put the French team back in front.
Alex Tulou then scored to give Castres a bonus point, before Meyer added a consolation penalty late on for Newport.
The Dragons remain top of Pool 2 in the Challenge Cup, while Castres move up into second.