diff --git a/doc/source/whatsnew/v0.20.0.txt b/doc/source/whatsnew/v0.20.0.txt
index 02c54f28a1695d..185d6b7d3ae83e 100644
--- a/doc/source/whatsnew/v0.20.0.txt
+++ b/doc/source/whatsnew/v0.20.0.txt
@@ -1643,6 +1643,7 @@ Plotting
 - Bug in ``DataFrame.boxplot`` where ``fontsize`` was not applied to the tick labels on both axes (:issue:`15108`)
 - Bug in the date and time converters pandas registers with matplotlib not handling multiple dimensions (:issue:`16026`)
 - Bug in ``pd.scatter_matrix()`` could accept either ``color`` or ``c``, but not both (:issue:`14855`)
+- Bug in ``plot`` where ``NaT`` in ``DatetimeIndex`` results in ``Timestamp.min`` (:issue: `12405`)
 
 Groupby/Resample/Rolling
 ^^^^^^^^^^^^^^^^^^^^^^^^
diff --git a/pandas/plotting/_core.py b/pandas/plotting/_core.py
index c3476d1443fc3a..389cd1c926ab35 100644
--- a/pandas/plotting/_core.py
+++ b/pandas/plotting/_core.py
@@ -11,6 +11,7 @@
 
 from pandas.util.decorators import cache_readonly
 from pandas.core.base import PandasObject
+from pandas.core.dtypes.missing import notnull
 from pandas.core.dtypes.common import (
     is_list_like,
     is_integer,
@@ -537,6 +538,7 @@ def _get_xticks(self, convert_period=False):
                 """
                 x = index._mpl_repr()
             elif is_datetype:
+                self.data = self.data[notnull(self.data.index)]
                 self.data = self.data.sort_index()
                 x = self.data.index._mpl_repr()
             else:
diff --git a/pandas/tests/plotting/test_datetimelike.py b/pandas/tests/plotting/test_datetimelike.py
index 803907c60d0d0b..be30690c4f5d61 100644
--- a/pandas/tests/plotting/test_datetimelike.py
+++ b/pandas/tests/plotting/test_datetimelike.py
@@ -6,7 +6,7 @@
 from pandas.compat import lrange, zip
 
 import numpy as np
-from pandas import Index, Series, DataFrame
+from pandas import Index, Series, DataFrame, NaT
 from pandas.compat import is_platform_mac
 from pandas.core.indexes.datetimes import date_range, bdate_range
 from pandas.core.indexes.timedeltas import timedelta_range
@@ -815,6 +815,23 @@ def test_mixed_freq_shared_ax(self):
         # self.assertEqual(ax1.lines[0].get_xydata()[0, 0],
         #                  ax2.lines[0].get_xydata()[0, 0])
 
+    def test_nat_handling(self):
+
+        import matplotlib.pyplot as plt  # noqa
+
+        fig = plt.gcf()
+        plt.clf()
+        ax = fig.add_subplot(111)
+
+        dti = DatetimeIndex(['2015-01-01', NaT, '2015-01-03'])
+        s = Series(range(len(dti)), dti)
+        s.plot(ax=ax)
+        xdata = ax.get_lines()[0].get_xdata()
+        # plot x data is bounded by index values
+        self.assertLessEqual(s.index.min(), Series(xdata).min())
+        self.assertLessEqual(Series(xdata).max(), s.index.max())
+        _check_plot_works(s.plot)
+
     @slow
     def test_to_weekly_resampling(self):
         idxh = date_range('1/1/1999', periods=52, freq='W')