And they managed to do just that, with Alex Cuthbert's second-half try proving the difference in a 16-5 win at Rodney Parade.
Steffan Jones' first-half try was the Dragons' only score and completing the scoring for the Blues was Rhys Patchell who added a conversion and three penalties.
"This derby is always so tough and straight away we knew this was going to be one of the hardest games of the season," said Wales captain Warburton.
"Dragons definitely didn't fail to deliver on that and they made it very tough and we had to work so hard.
"I am absolutely delighted with the performance. Last week we were so disappointed to lose to that drop-goal to Edinburgh in the last five minutes, and we really wanted to get a win (at Newport).
"The first ten minutes of the second half were really good for us. We managed to have some decent territory and we came out on top points wise."
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