For positive-sense RNA viruses, initiation of viral RNA replication represents a major target of antiviral responses to infection. Despite this, the interplay between viral replication and the innate antiviral response at early steps in the Zika virus (ZIKV) life cycle is not well understood. We have previously identified ZIKV isolates with differing levels of dsRNA accumulation, ZIKVPR (high dsRNA per infected cell) and ZIKVCDN (low dsRNA per infected cell), and we hypothesized that we could use reverse genetics to investigate how host and viral factors contribute to the establishment of viral RNA replication. We found that both the ZIKV NS3 and NS5 proteins as well as host factors were necessary to determine the dsRNA accumulation phenotype. Additionally, we show that dsRNA correlates with viral negative-strand RNA measured by strand-specific RT-qPCR, suggesting that dsRNA is an accurate readout of viral RNA replication. Interestingly, although we did not observe NS3- and NS5-dependent differences in cells with defects in interferon (IFN) production, differences in RNA accumulation precede induction of the IFN response, suggesting that RNA sensing pathways or intrinsic restriction factors may differentially restrict ZIKV in an NS3- and NS5-dependent manner. This work expands our understanding of the interplay of early steps of viral RNA replication and the induction of the innate antiviral response to ZIKV infection.