The present paper is devoted to the well-posedness of a type of multi-dimensional backward stochastic differential equations (BSDEs) with a diagonally quadratic generator. We give a new priori estimate, and prove that the BSDE admits a unique solution on a given interval when the generator has a sufficiently small growth of the off-diagonal elements (i.e., for each $i$, the $i$-th component of the generator has a small growth of the $j$-th row $z^j$ of the variable $z$ for each $j \neq i$). Finally, we give a solvability result when the diagonally quadratic generator is triangular.