Many protostellar gapped and binary discs show misalignments between their inner and outer discs; in some cases, ∼70° misalignments have been observed. Here, we show that these misalignments can be generated through a secular resonance between the nodal precession of the inner disc and the precession of the gap-opening (stellar or massive planetary) companion. An evolving protostellar system may naturally cross this resonance during its lifetime due to disc dissipation and/or companion migration. If resonance crossing occurs on the right time-scale, of the order of a few million years, characteristic for young protostellar systems, the inner and outer discs can become highly misaligned, with misalignments ≳ 60° typical. When the primary star has a mass of order a solar mass, generating a significant misalignment typically requires the companion to have a mass of ∼0.01–0.1 M⊙ and an orbital separation of tens of astronomical units. The recently observed companion in the cavity of the gapped, highly misaligned system HD 142527 satisfies these requirements, indicating that a previous resonance crossing event misaligned the inner and outer discs. Our scenario for HD 142527's misaligned discs predicts that the companion's orbital plane is aligned with the outer disc's; this prediction should be testable with future observations as the companion's orbit is mapped out. Misalignments observed in several other gapped disc systems could be generated by the same secular resonance mechanism.