We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDEs). This class is characterized by constraints on some uniform a priori estimates on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921–2939] concerning scalar quadratic BSDEs.