Striker Palmer made the most of his recall by converting Rob Taylor's cross just after the hour mark to ensure manager-less Mansfield booked a trip to League Two rivals Cambridge United in the second round on December 6.
Conference South outfit Rangers had secured a 1-1 draw at Field Mill to earn a first-round replay, but were unable to cause a shock and Mansfield could have won by a more emphatic margin.
Substitute Fergus Bell was denied by the post in the second half and Palmer was denied a second goal by Rangers goalkeeper Josh Vickers, but the missed chance did not prove costly.