Database fingerprinting schemes have been widely adopted to prevent unauthorized sharing of data and identify the source of data leakages. Although existing schemes are robust against common attacks, such as random bit flipping and subset attack, their robustness degrades significantly if attackers utilize the inherent correlations among database entries. In this paper, we first demonstrate this vulnerability of the existing database fingerprinting schemes by identifying different correlation attacks: column-wise correlation attack, row-wise correlation attack, and the integration of them. To provide robust fingerprinting against the identified correlation attacks, we then develop mitigation techniques, which can work as post-processing steps for any off-the-shelf database fingerprinting schemes. The proposed mitigation techniques also preserve the utility of the fingerprinted database considering different utility metrics. We empirically investigate the impact of the identified correlation attacks and the performance of mitigation techniques using two real-world relational databases. Our results show (i) high success rates of the identified correlation attacks against existing fingerprinting schemes (e.g., the integrated correlation attack can distort 64.8% fingerprint bits by just modifying 14.2% entries in a fingerprinted database), and (ii) high robustness of the proposed mitigation techniques (e.g., after the proposed mitigation techniques, the integrated correlation attack can only distort 3% fingerprint bits). Furthermore, we show that the proposed mitigation techniques effectively alleviate correlation attacks even if the database owner has less accurate knowledge about data correlations compared to the attacker.