Separable　nonlinear　models　(SNLMs)　are　of　great　importance　in　system　modeling,　signal　processing,　and　machine　learning　because　of　their　flexible　structure　and　excellent　description　of　nonlinear　behaviors.　The　online　identification　of　such　models　is　quite　challenging,　and　previous　related　work　usually　ignores　the　special　structure　where　the　estimated　parameters　can　be　partitioned　into　a　linear　and　a　nonlinear　part.　In　this　brief,　we　propose　an　efficient　first-order　recursive　algorithm　for　SNLMs　by　introducing　the　variable　projection　(VP)　step.　The　proposed　algorithm　utilizes　the　recursive　least-squares　method　to　eliminate　the　linear　parameters,　resulting　in　a　reduced　function.　Then,　the　stochastic　gradient　descent　(SGD)　algorithm　is　employed　to　update　the　parameters　of　the　reduced　function.　By　considering　the　tight　coupling　relationship　between　linear　parameters　and　nonlinear　parameters,　the　proposed　first-order　VP　algorithm　is　more　efficient　and　robust　than　the　traditional　SGD　algorithm　and　alternating　optimization　algorithm.　More　importantly,　since　the　proposed　algorithm　just　uses　the　first-order　information,　it　is　easier　to　apply　it　to　large-scale　models.　Numerical　results　on　examples　of　different　sizes　confirm　the　effectiveness　and　efficiency　of　the　proposed　algorithm.