We have studied the low-temperature magnetoresistance of Y_3/4Lu_1/4Ba₂Cu₃O₇-CuO composites obtained by fast sintering technique and established a relation between the probing to critical current density ratio j/j _c and the shape of the magnetoresistance curve ρ(H). For j/j _c<1, the electric resistance arises at a threshold value of the magnetic field strength H _c. For j/j _c≥1, a linear variation of ρ(H) at 77 K in the range from 0 to 14 Oe can be provided by selecting the CuO content (in the 15–30 vol % interval) and the j value (in the 0.003–0.2 A/cm² range). In the latter case, the slope dρ/dH (i.e., the sensitivity of the electric resistivity with respect to the magnetic field) is 1–20 mΩ cm/Oe and the relative field-induced increase in the resistivity ρ₀=(ρ(H)−ρ(H=0))/ρ(H=0) amounts to 1320 and 685% at H=200 and 35 Oe, respectively. Composites possessing controlled magnetoresistance are promising materials for the active elements of magnetic field sensors capable of operating at a practically convenient liquid nitrogen temperature.