This paper proposes a general analytical approach to derive the outage probability of hybrid automatic repeat request with incremental redundancy (HARQ-IR) over correlated fading channels in closed-form. Unlike prior analyses, the consideration of channel correlation is one of the key reasons making the outage analysis involved. Using conditional Mellin transform, the outage probability of HARQ-IR over correlated Rayleigh fading channels is exactly expressed as a mixture of outage probabilities of HARQ-IR over independent Nakagami fading channels, where the weights are negative multinomial probabilities. Its straightforward application is to conduct asymptotic outage analysis to gain more insights, in which the asymptotic outage probability is obtained in a concise form. The asymptotic outage probability possesses some special properties which ease optimal power allocation and rate selection of HARQ-IR. Finally, numerical results are presented for validations and discussions.