In order to prevent unwanted access to sensitive data by unauthorized individuals, color images are encoded. Because chaotic-deoxyribonucleic acid (chaotic-DNA) encoding can make information highly secure, it is often employed in image encryption. In this study, a new image encryption technique has been proposed based on a new 4D-chaotic system and DNA computing. The algorithm consists of two phases: in the first phase, the pixel positions are permuted by chaotic sequences. In the second phase, according to the concept of DNA cryptography, a set of operations (like DNA addition, DNA XOR, DNA subtraction, shift right, and shift left) are performed on the DNA encoding sequence. The performance of the suggested algorithm is evaluated through analyses like correlation coefficient, entropy, histogram, and key space. The results show that the encryption method that was exhibited has good encryption performance and high security. For encrypted images, the histogram is fairly uniform, the correlation values between adjacent pixels are very small and close to zero, and the entropy is near to the ideal value of eight. In addition, the proposed system has a very large key space that is equal to 2627 keys, which makes it resistant to brute-force, differential, and statistical attacks.