In the current scenario ensuring data security has become an essential aspect due to the rise in data movement in public networks. The traditional methods of cryptography are now replaced by the quicker and more efficient chaotic maps which are being used in this data security aspect. The Tinkerbell 2D cat map is a common instance among the chaotic maps. In this article a completely new method of Image encryption and decryption employing a three-dimensional variant of the classical Tinkerbell chaotic map is proposed. The third dimension provides a more defined window of perception which is intriguing and secure. The capability of the suggested approach is surveyed thoroughly in terms of both colors and monochromatic images and its performative appraisal has been explored on the basis of several indices such as NPCR, PSNR, UACI etc. It is observed that the proposed 3D extension of Tinkerbell chaotic map has provided superior results than the older versions in terms of the performance-evaluation matrices. The results are similar for MSE, PSNR and NCC measures. Histogram analysis of original image, encrypted image, and decrypted image verify the results and concludes that a 3D chaotic map provides additional security as the images encrypted in the example had very uniform histograms. The applicability of the 3D chaotic map to encrypt and decrypt image is confirmed clearly in this research through various color and grayscale images tested using the proposed map.