Many optical systems require broadband filters with sharp roll-offs for efficiently splitting or combining light across wide spectra. While free space dichroic filters can provide broadband selectivity, on-chip integration of these high-performance filters is crucial for the scalability of photonic applications in multi-octave interferometry, spectroscopy, and wideband wavelength-division multiplexing. Here we present the theory, design, and experimental characterization of integrated, transmissive, 1 × 2 port dichroic filters using spectrally selective waveguides. Mode evolution through adiabatic transitions in the demonstrated filters allows for single cutoff and flat-top responses with low insertion losses and octave-wide simulated bandwidths. Filters with cutoffs around 1550 and 2100 nm are fabricated on a silicon-on-insulator platform with standard complementary metal-oxide-semiconductor processes. A filter roll-off of 2.82 dB nm-1 is achieved while maintaining ultra-broadband operation. This new class of nanophotonic dichroic filters can lead to new paradigms in on-chip communications, sensing, imaging, optical synthesis, and display applications.