We demonstrate a terahertz detection technique based on variations of the energy of femtosecond laser pulses with an edge-cut spectrum in birefringent electro-optic crystal-periodically poled lithium niobate (PPLN). The method is compared with the standard electro-optic detection scheme utilizing GaP crystal. The experimental results show that the studied technique is suited for use with birefringent crystals and allows one to achieve a much better response to the terahertz wave radiation at quasi-phase-matching frequencies of PPLN.