diff --git a/pandas/core/arrays/sparse/array.py b/pandas/core/arrays/sparse/array.py
index 4847372f18239..6ab296b314615 100644
--- a/pandas/core/arrays/sparse/array.py
+++ b/pandas/core/arrays/sparse/array.py
@@ -1397,7 +1397,7 @@ def max(self, axis=0, *args, **kwargs):
 
         # This condition returns a nan if there are no valid values in the array.
         if self.size > 0 and self._valid_sp_values.size == 0:
-            return np.nan
+            return self.fill_value
         else:
             return np.nanmax(self, axis)
 
@@ -1406,7 +1406,7 @@ def min(self, axis=0, *args, **kwargs):
 
         # This condition returns a nan if there are no valid values in the array.
         if self.size > 0 and self._valid_sp_values.size == 0:
-            return np.nan
+            return self.fill_value
         else:
             return np.nanmin(self, axis)
 
diff --git a/pandas/tests/arrays/sparse/test_array.py b/pandas/tests/arrays/sparse/test_array.py
index b29855caf6c1d..1cc8a2df44812 100644
--- a/pandas/tests/arrays/sparse/test_array.py
+++ b/pandas/tests/arrays/sparse/test_array.py
@@ -1326,6 +1326,9 @@ class TestMinMax:
     data_neg = plain_data * (-1)
     data_NaN = SparseArray(np.array([0, 1, 2, np.nan, 4]))
     data_all_NaN = SparseArray(np.array([np.nan, np.nan, np.nan, np.nan, np.nan]))
+    data_NA_filled = SparseArray(
+        np.array([np.nan, np.nan, np.nan, np.nan, np.nan]), fill_value=5
+    )
 
     @pytest.mark.parametrize(
         "raw_data,max_expected,min_expected",
@@ -1334,6 +1337,7 @@ class TestMinMax:
             (data_neg, [0], [-4]),
             (data_NaN, [4], [0]),
             (data_all_NaN, [np.nan], [np.nan]),
+            (data_NA_filled, [5], [5]),
         ],
     )
     def test_maxmin(self, raw_data, max_expected, min_expected):