Several gamma-ray observations from distant blazars show a suppressed emission of the inverse Compton scattering cascade of the blazar-induced pair beams at the GeV energy band. There are two possible explanations, the first one is the deflections of the pair beam electrons and positrons by magnetic fields in the intergalactic medium. The second one is the drain of the pair energy by beam-plasma instabilities resulting in heating up the intergalactic plasma. The studies of plasma instabilities of blazar-induced pair beams in the literature neglect the effect of weak intergalactic magnetic fields. In this work, we investigate the effect of weak intergalactic magnetic fields with small correlation lengths on the electrostatic beam-plasma instability. Such weak fields do not modify the dispersion relation describing the electrostatic waves and so the linear growth rate description. However, We found that the increase of the angular distribution of the particles of the pair beam due to the weak intergalactic fields reduces the linear growth rate of the electrostatic instability. This reduction of the linear growth rate increases the energy loss time of the beam-plasma instability suppressing it against the inverse Compton scattering at magnetic field strengths three orders of magnitude less than the lower limit of the magnetic fields needed to deflect the secondary cascade emission.