The $^{74}$Ge(p,γ)$^{75}$As reaction plays a crucial role in producing the lightest p-nucleus $^{74}$Se. In this work, we present the measurement of the $^{74}$Ge(p,γ)$^{75}$As cross sections in the range of Ep = 1.4-2.8 MeV with an in-beam method at the tandem accelerator of China Institute of Atomic Energy. The present result is in good agreement with the previous experiments at higher energies, and extends the measurement down to lower energies of astrophysical interest. Theoretical calculations are also performed using the EMPIRE nuclear reaction code and well reproduce all experimental cross sections, which shed some light on the existing discrepancy in the previously reported experimental and theoretical values at the lowest energies. The resulting reaction rates exhibit a considerable increase of up to 47% within the p-process relevant temperatures when comparing with the REACLIB compilation.
