diff --git a/doc/source/whatsnew/v0.24.0.rst b/doc/source/whatsnew/v0.24.0.rst
index 5213120b33f06..bcd1abba30a67 100644
--- a/doc/source/whatsnew/v0.24.0.rst
+++ b/doc/source/whatsnew/v0.24.0.rst
@@ -1808,6 +1808,7 @@ Plotting
 Groupby/Resample/Rolling
 ^^^^^^^^^^^^^^^^^^^^^^^^
 
+- Bug in :func:`pandas.core.window.Rolling.min` and :func:`pandas.core.window.Rolling.max` with ``closed='left'``, a datetime-like index and only one entry in the series leading to segfault (:issue:`24718`)
 - Bug in :func:`pandas.core.groupby.GroupBy.first` and :func:`pandas.core.groupby.GroupBy.last` with ``as_index=False`` leading to the loss of timezone information (:issue:`15884`)
 - Bug in :meth:`DateFrame.resample` when downsampling across a DST boundary (:issue:`8531`)
 - Bug in date anchoring for :meth:`DateFrame.resample` with offset :class:`Day` when n > 1 (:issue:`24127`)
diff --git a/pandas/_libs/window.pyx b/pandas/_libs/window.pyx
index 5f2c0233a0f13..e8f3de64c3823 100644
--- a/pandas/_libs/window.pyx
+++ b/pandas/_libs/window.pyx
@@ -1339,7 +1339,10 @@ cdef _roll_min_max_variable(ndarray[numeric] values,
             Q.push_back(i)
             W.push_back(i)
 
-        output[N-1] = calc_mm(minp, nobs, values[Q.front()])
+        if not Q.empty():
+            output[N-1] = calc_mm(minp, nobs, values[Q.front()])
+        else:
+            output[N-1] = NaN
 
     return output
 
diff --git a/pandas/tests/test_window.py b/pandas/tests/test_window.py
index 412f70a3cb516..e816d4c04344a 100644
--- a/pandas/tests/test_window.py
+++ b/pandas/tests/test_window.py
@@ -520,6 +520,26 @@ def test_closed(self):
         with pytest.raises(ValueError):
             df.rolling(window=3, closed='neither')
 
+    @pytest.mark.parametrize("func", ['min', 'max'])
+    def test_closed_one_entry(self, func):
+        # GH24718
+        ser = pd.Series(data=[2], index=pd.date_range('2000', periods=1))
+        result = getattr(ser.rolling('10D', closed='left'), func)()
+        tm.assert_series_equal(result, pd.Series([np.nan], index=ser.index))
+
+    @pytest.mark.parametrize("func", ['min', 'max'])
+    def test_closed_one_entry_groupby(self, func):
+        # GH24718
+        ser = pd.DataFrame(data={'A': [1, 1, 2], 'B': [3, 2, 1]},
+                           index=pd.date_range('2000', periods=3))
+        result = getattr(
+            ser.groupby('A', sort=False)['B'].rolling('10D', closed='left'),
+            func)()
+        exp_idx = pd.MultiIndex.from_arrays(arrays=[[1, 1, 2], ser.index],
+                                            names=('A', None))
+        expected = pd.Series(data=[np.nan, 3, np.nan], index=exp_idx, name='B')
+        tm.assert_series_equal(result, expected)
+
     @pytest.mark.parametrize("input_dtype", ['int', 'float'])
     @pytest.mark.parametrize("func,closed,expected", [
         ('min', 'right', [0.0, 0, 0, 1, 2, 3, 4, 5, 6, 7]),