This paper presents a new algorithm for computing correctly rounded dot products using floating-point (FP) operations under the round-to-odd (RO) rounding mode. Our algorithm applies error-free transformations (EFT) for FP multiplication to reduce dot products to summations over vectors. By leveraging EFTs for FP addition tailored to RO, we implement an algorithm that produces faithfully rounded sums over vectors under RO. Using the faithfully rounded sum and the properties of RO, our dot product algorithm subsequently determines the correctly rounded result. Through correct rounding, our algorithm enables accurate dot products and matrix multiplications for future hardware that natively supports RO.