Dual Fibonacci and dual Lucas numbers are defined with dual Fibonacci and Lucas quaternions in Nurkan and Guven [14]. In this study, we define the dual third-order Jacobsthal quaternion and the dual third-order Jacobsthal-Lucas quaternion. We derive the relations between the dual third-order Jacobsthal quaternion and dual third-order Jacobsthal-Lucas quaternion which connected the third-order Jacobsthal and third-order Jacobsthal-Lucas numbers. In addition, we give the generating functions, the Binet and Cassini formulas for these new types of quaternions.