Gravitational-wave memory is a low-frequency, non-oscillatory signal that provides a promising probe of strong-field gravity. We present the first computation of memory from full inspiral--merger--ringdown waveforms in a theory beyond GR, focusing on scalar Gauss--Bonnet gravity. We find percent-level deviations from GR, mainly driven by modified merger dynamics, while scalar-induced contributions to tensor memory are strongly suppressed. We found that including memory greatly enhances the mismatch between GR and beyond-GR waveforms, highlighting its potential as a complementary observable for tests of gravity with next-generation detectors.