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The Atlantic Meridional Overturning Circulation (AMOC) impacts temperatures, ecosystems, and the carbon cycle. However, AMOC effects on Earth's carbon cycle remains poorly understood, in part because contributions of different physical and biological mechanisms that impact carbon storage in the ocean are not typically diagnosed in climate models. Here, we explore modeled effects of AMOC shutdowns on ocean Dissolved Inorganic Carbon (DIC) by applying a new decomposition that explicitly calculates preformed and regenerated DIC components and separates physical and biological contributions. An extensive evaluation in transient simulations finds that the method is accurate, especially for basin‐wide changes, whereas errors can be significant at global and local scales. In contrast, estimates of respired carbon based on Apparent Oxygen Utilization lead to large errors and are generally not reliable. In response to a shutdown of the AMOC under Last Glacial Maximum (LGM) background climate, ocean carbon increases and then decreases, leading to opposite changes in atmospheric carbon dioxide (CO2). DIC changes are dominated by opposing changes in biological carbon storage. Whereas regenerated components increase in the Atlantic and dominate the initial increase in global ocean DIC until model year 1000, preformed components decrease in the other ocean basins and dominate the long‐term DIC decrease until year 4000. Biological disequilibrium is an important contribution to preformed carbon changes. Biological saturation carbon decreases in the Pacific, Indian, and Southern Oceans due to a decrease in surface alkalinity. The spatial patterns of the DIC components and their changes in response to an AMOC collapse are presented.