A powertrain control apparatus that controls a powertrain including a lock-up clutch, which connects an engine, in which fuel supply may be cut off, directly to an automatic transmission. The control apparatus includes an output unit, a control unit, and a setting unit. The output unit outputs an instruction to lower engagement pressure for the lock-up clutch from a value at which the lock-up clutch is engaged to a value at which the lock-up clutch is disengaged. The control unit resumes the fuel supply to the engine when a predetermined lag time has elapsed since the instruction is output. The setting unit sets the lag time so that the lag time is shorter when a temperature of a combustion chamber of the engine is a first temperature than when the temperature of the combustion chamber of the engine is a second temperature which is higher than the first temperature.