The equivalent of the overall F-Test in logistic regression is Deviance.

The difference between the null and residual deviance can be approximated by a Chi-Squared distribution with degrees of freedom equal to the difference in number of parameters between the proposed and null model using a likelihood ratio test. With a Chi-squared test statistic and corresponding degrees of freedom, a p-value can be obtained. If the p-value is small, there is significant evidence that the proposed model is doing a better job than a model with no predictors in fitting the data.