This paper analyzes Type I hybrid automatic repeat request (HARQ) over time-correlated Rayleigh fading channels. Due to the presence of channel time correlation, the analysis is more challenging than the prior analysis in the literature. Outage probability is first derived as a weighted sum of joint CDF of multiple independent Gamma random variables based on an infinite series representation. A truncation method is proposed for efficient computation of the outage probability and it is proved that the truncation error decreases exponentially with the truncation order. Asymptotic outage probability is then derived in a simple form, with which the impacts of packet transmission rate, transmit power and channel time correlation could be decoupled and analyzed clearly. Based on the asymptotic outage probability, diversity order of HARQ is also analyzed. It is found that full diversity can be achieved even under time correlated fading channels and the time correlation of the channels has negative effect on the outage probability under high signal-to-noise ratio (SNR). Finally, our analytical results are validated by Monte-Carlo simulations.