Mason and Remmel introduced a basis for quasisymmetric functions known as the row-strict quasisymmetric Schur functions. This basis is generated combinatorially by fillings of composition diagrams that are analogous to the row-strict tableaux that generate Schur functions. We introduce a modification known as Young row-strict quasisymmetric Schur functions, which are generated by row-strict Young composition fillings. After discussing basic combinatorial properties of these functions, we define a skew Young row-strict quasisymmetric Schur function using the Hopf algebra of quasisymmetric functions and then prove this is equivalent to a combinatorial description. We also provide a decomposition of the skew Young row-strict quasisymmetric Schur functions into a sum of Gessel's fundamental quasisymmetric functions and prove a multiplication rule for the product of a Young row-strict quasisymmetric Schur function and a Schur function.