We use a VAR model with two exogenous and eight endogenous variables to evaluate the asymmetric effect of oil price changes on the EUR/USD exchange rate. The parameters are estimated with the Bayesian technique using the U.S. and the Eurozone quarterly data from 2001 to 2021. The sign restriction methodology is employed to obtain the VAR model’s structural form from its reduced form. The suggested metrics show highly time-variant oil price effects on the EUR/USD exchange rate. Oil price variability accounts for about half of the exogenous variables shocks that account for about one-third of the deterministic part of the model, which explains about two-thirds of the EUR/USD exchange rate volatility. The structural shock and historical variance decomposition identify significant time-dependent endogenous variables shocks behind the EUR/USD exchange rate variability. The impulse–response function shows asymmetric direct and indirect effects of oil shocks on the EUR/USD exchange rate, whose reaction to a positive oil price shock is more intensive, less smooth, and takes longer than in a negative shock. The dollar appreciates within three quarters after oil price increases, which suggests the United States has a specific comparative advantage compared with the Eurozone in offsetting the effects of the oil shocks.