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Automated bidding to optimize online advertising with various constraints, e.g. ROI constraints and budget constraints, is widely adopted by advertisers. A key challenge lies in designing algorithms for non-truthful mechanisms with ROI constraints. While prior work has addressed truthful auctions or non-truthful auctions with weaker benchmarks, this paper provides a significant improvement: We develop online bidding algorithms for repeated first-price auctions with ROI constraints, benchmarking against the optimal randomized strategy in hindsight. In the full feedback setting, where the maximum competing bid is observed, our algorithm achieves a near-optimal $\tilde O(\sqrt{T})$ regret bound, and in the bandit feedback setting (where the bidder only observes whether the bidder wins each auction), our algorithm attains $\tilde O(T^{3/4})$ regret bound.