We consider the Bayesian binary regression model and we introduce a new class of distributions, the Perturbed Unified Skew-Normal (pSUN, henceforth), which generalizes the Unified Skew-Normal (SUN) class. We show that the new class is conjugate to any binary regression model, provided that the link function may be expressed as a scale mixture of Gaussian CDFs. We discuss in detail the popular logit case, and we show that, when a logistic regression model is combined with a Gaussian prior, posterior summaries such as cumulants and normalizing constants can easily be obtained through the use of an importance sampling approach, opening the way to straightforward variable selection procedures. For more general prior distributions, the proposed methodology is based on a simple Gibbs sampler algorithm. We also claim that, in the p>n case, our proposal presents better performances - both in terms of mixing and accuracy - compared to the existing methods.