diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index 57e3c9dd66afb..1336fd7d83f7e 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -476,6 +476,7 @@ Other
 - Bug in :meth:`DataFrame.replace` and :meth:`Series.replace` with numeric values and string ``to_replace`` (:issue:`34789`)
 - Fixed metadata propagation in the :class:`Series.dt` accessor (:issue:`28283`)
 - Bug in :meth:`Index.union` behaving differently depending on whether operand is a :class:`Index` or other list-like (:issue:`36384`)
+- Passing an array with 2 or more dimensions to the :class:`Series` constructor now raises the more specific ``ValueError``, from a bare ``Exception`` previously (:issue:`35744`)
 
 .. ---------------------------------------------------------------------------
 
diff --git a/pandas/core/construction.py b/pandas/core/construction.py
index 4751f6076f869..7901e150a7ff4 100644
--- a/pandas/core/construction.py
+++ b/pandas/core/construction.py
@@ -510,7 +510,7 @@ def sanitize_array(
 
     elif subarr.ndim > 1:
         if isinstance(data, np.ndarray):
-            raise Exception("Data must be 1-dimensional")
+            raise ValueError("Data must be 1-dimensional")
         else:
             subarr = com.asarray_tuplesafe(data, dtype=dtype)
 
diff --git a/pandas/tests/series/test_constructors.py b/pandas/tests/series/test_constructors.py
index 4ad4917533422..a950ca78fc742 100644
--- a/pandas/tests/series/test_constructors.py
+++ b/pandas/tests/series/test_constructors.py
@@ -113,8 +113,8 @@ def test_constructor(self, datetime_series):
         with tm.assert_produces_warning(DeprecationWarning, check_stacklevel=False):
             assert not Series().index._is_all_dates
 
-        # exception raised is of type Exception
-        with pytest.raises(Exception, match="Data must be 1-dimensional"):
+        # exception raised is of type ValueError GH35744
+        with pytest.raises(ValueError, match="Data must be 1-dimensional"):
             Series(np.random.randn(3, 3), index=np.arange(3))
 
         mixed.name = "Series"