Enhanced evaporation from the Arctic, driven by ongoing sea-ice reduction, is increasingly contributing to atmospheric moisture and altering precipitation patterns around the Arctic. This added moisture, besides being shown to shift Arctic precipitation to be more locally sourced, can also have a broader impact on land precipitation. Using a moisture-tracking model, we established physical links between sea-ice loss in the Arctic marginal seas (AMS) and changes in AMS-sourced precipitation, distinguishing the contributions of evaporation and atmospheric circulation. Our analysis revealed that during the cold seasons of 1980-2021, there was a significant 16% increase in AMS-sourced land precipitation for each million square kilometers of sea ice lost. Despite being a minor component of land precipitation, the augmented AMS moisture disproportionately influenced changes in precipitation, which is particularly evident in high-latitude regions. The evolution of atmospheric circulation emerges as a crucial factor in shaping the patterns of the AMS moisture contribution over land, especially during extreme events. The additional moisture has fueled extreme snowfall, potentially contributing to glacier mass gain and helping mitigate the impact of climate warming on Greenland ice sheet melting. Our findings highlight the importance of the progressively ice-free Arctic as a critical contributor to the rising levels of cold-season precipitation and snowfall over northern high-latitude land.