Quasi-steady state time-series (QSTS) simulation of distribution feeders can become computationally burdensome due to many buses and devices, long simulation horizons, and/or high temporal resolution. To reduce this burden, network reduction removes buses and shifts loads/generation to the remaining buses of the circuit to produce a smaller equivalent. However, voltage-controlled devices have traditionally limited network reduction, since their operation depends on the measurement of voltage at their local bus. This work includes the reduction of buses with voltage-controlled devices by replacing the local voltage measurement with an estimate from a fast voltage sensitivity approach, which is integrated directly into a modified QSTS simulation. Comprehensive tests on an unbalanced feeder with real operating data and volt-var controlled inverters show agreement in cumulative reactive power output between the reduced and the original feeder circuits. The maximum voltage error is 0.005 Vp.u., which is nearly identical to the error in a benchmark reduction without smart inverter voltage control. The algorithm convergences for every time step, even when reducing the frequency of which the voltage estimation was updated. While the reduction methodology is demonstrated for inverter volt-var control, since it represents a frequent use case, it can be extended to other voltage-controlled devices.