In one aspect, a fuel supply system may include a fuel injector having a primary and a secondary pilot fuel nozzle in fluid communication with a primary and a secondary fuel circuit, respectively, and a main fuel nozzle in fluid communication with a main fuel circuit. The fuel injector may also define a by-pass fuel circuit connected between the primary circuit and the secondary circuit and/or the main circuit. The system may also include a primary fuel manifold configured to be fluidly connected to the primary pilot fuel nozzle via the primary fuel circuit. Moreover, the system may include a by-pass valve provided in operative association with the by-pass fuel circuit. The by-pass valve may be configured to be opened such that a portion of the fuel flowing through the primary circuit from the primary fuel manifold is directed to the secondary circuit and/or the main circuit.