You should be entitled to the arrears, which is the difference between the assessment rate, and the rate for the group in which you have been placed, ie the work-related activity or support group, from the 14th week of your ESA claim.
It may take a while to receive these arrears, but what the DWP is telling you is wrong, unless they intend to appeal to the Upper Tribunal on a point of law against the tribunal's decision, in which case you may not get paid the arrears until the appeal is decided?
I think it would be wise to contact the DWP office again that deals with your claim, send a copy of the decision notice in the post, and ask them if they intend to appeal the tribunal's decision? If not, they are legally obliged to issue the arrears to you without delay.
Nothing on this board constitutes legal advice - always consult a professional about specific problems