Individual signaling pathways, such as fibroblast growth factors (FGFs), can regulate a plethora of inductive events. According to current paradigms, signal-dependent transcription factors (TFs), such as FGF/MapK-activated Ets family factors, partner with lineage-determining factors to achieve regulatory specificity. However, many aspects of this model have not been rigorously investigated. One key question relates to whether lineage-determining factors dictate lineage-specific responses to inductive signals or facilitate these responses in collaboration with other inputs. We utilize the chordate model Ciona robusta to investigate mechanisms generating lineage-specific induction. Previous studies in C. robusta have shown that cardiopharyngeal progenitor cells are specified through the combined activity of FGF-activated Ets1/2.b and an inferred ATTA-binding transcriptional cofactor. Here, we show that the homeobox TF Lhx3/4 serves as the lineage-determining TF that dictates cardiopharyngeal-specific transcription in response to pleiotropic FGF signaling. Targeted knockdown of Lhx3/4 leads to loss of cardiopharyngeal gene expression. Strikingly, ectopic expression of Lhx3/4 in a neuroectodermal lineage subject to FGF-dependent specification leads to ectopic cardiopharyngeal gene expression in this lineage. Furthermore, ectopic Lhx3/4 expression disrupts neural plate morphogenesis, generating aberrant cell behaviors associated with execution of incompatible morphogenetic programs. Based on these findings, we propose that combinatorial regulation by signal-dependent and lineage-determinant factors represents a generalizable, previously uncategorized regulatory subcircuit we term "cofactor-dependent induction." Integration of this subcircuit into theoretical models will facilitate accurate predictions regarding the impact of gene regulatory network rewiring on evolutionary diversification and disease ontogeny.