We demonstrate resonance fluorescence from single In-GaAs/GaAs quantum dots embedded in a rib waveguide beamsplitter structure operated under pulsed laser excitation. A systematic study on the excitation laser pulse duration depicts that a sufficiently small laser linewidth enables a substantial improved single-photon-to-laser-background ratio inside a waveguide chip. This manifests in the observation of clear Rabi oscillations over two periods of the quantum dot emission as a function of laser excitation power. A photon cross-correlation measurement between the two output arms of an on-chip beamsplitter results in a g(2)(0)=0.18, demonstrating the generation, guiding and splitting of triggered single photons under resonant excitation in an on-chip device. The present results open new perspectives for the implementation of photonic quantum circuits with integrated quantum dots as resonantly-pumped deterministic single-photon sources.