The　evaluation　of　the　first-passage　probability　of　non-stationary　non-Gaussian　structural　responses　remains　a　great　challenge　in　the　field　of　random　vibrations.　In　the　present　paper,　a　novel　method　is　proposed　for　evaluating　this　first-passage　probability,　whose　main　contribution　is　to　construct　the　joint　probability　density　function　(PDF)　of　the　structural　response　and　its　derivative　process　under　the　consideration　of　their　non-Gaussianities　and　nonlinear　correlations.　Cubic　polynomial　models　of　Gaussian　process　are　developed　to　characterize　the　non-Gaussianities　of　the　structural　response　and　its　derivative　process,　whose　polynomial　coefficients　at　each　instant　time　are　explicitly　determined　from　their　corresponding　first　four　linear　moments.　These　linear　moments　are　accurately　evaluated　using　a　proposed　method　combining　Sobol　sequence　with　polynomial　smoothing.　The　marginal　PDFs　and　cumulative　distribution　functions　(CDFs)　of　the　structural　response　and　its　derivative　process　are　then　derived　from　these　polynomial　models.　Based　on　Akaike　information　criterion　(AIC)　and　the　marginal　CDFs,　the　optimal　copula　function　at　each　instant　time　is　selected　to　capture　the　linear/nonlinear　correlation　between　the　structural　response　and　its　derivative　process.　And　thus,　the　joint　PDF　is　constructed　and　the　first-passage　probability　is　evaluated.　The　applicability　of　the　proposed　method　is　validated　by　several　numerical　examples.　It　can　be　concluded　that　the　proposed　method　provides　satisfactory　results　in　evaluating　the　linear　moments,　fitting　probability　distributions,　and　estimating　the　first-passage　probabilities　of　non-stationary　non-Gaussian　structural　responses.