diff --git a/python/pyarrow/array.pxi b/python/pyarrow/array.pxi
index bf9a6d67228..f460ebcfff7 100644
--- a/python/pyarrow/array.pxi
+++ b/python/pyarrow/array.pxi
@@ -2390,12 +2390,15 @@ cdef class Array(_PandasConvertible):
                 )
             # Note: from March 2025 onwards, it's okay to raise BufferError here.
             # Still we keep the V0 version as the V1 was only added in August 2026.
+            legacy_tensor = GetResultValue(ExportArrayToDLPack(self.sp_array))
+            # Wrap the exported tensor in its capsule before warning so that a
+            # warning escalated to an error cannot leak the tensor.
+            capsule = PyCapsule_New(legacy_tensor, 'dltensor', dlpack_pycapsule_deleter)
             warnings.warn(
                 "Exporting an unversioned DLPack capsule is deprecated, "
                 "pass max_version=(1, 0) or higher.",
                 DeprecationWarning, stacklevel=2)
-            legacy_tensor = GetResultValue(ExportArrayToDLPack(self.sp_array))
-            return PyCapsule_New(legacy_tensor, 'dltensor', dlpack_pycapsule_deleter)
+            return capsule
 
         # Currently no major version other than legacy 0 and current 1.3
         dlm_tensor = GetResultValue(
diff --git a/python/pyarrow/tensor.pxi b/python/pyarrow/tensor.pxi
index 7f006f29417..c66fc6321de 100644
--- a/python/pyarrow/tensor.pxi
+++ b/python/pyarrow/tensor.pxi
@@ -401,12 +401,15 @@ strides: {self.strides}"""
                 )
             # Note: from March 2025 onwards, it's okay to raise BufferError here.
             # Still we keep the V0 version as the V1 was only added in August 2026.
+            legacy_tensor = GetResultValue(ExportTensorToDLPack(self.sp_tensor))
+            # Wrap the exported tensor in its capsule before warning so that a
+            # warning escalated to an error cannot leak the tensor.
+            capsule = PyCapsule_New(legacy_tensor, 'dltensor', dlpack_pycapsule_deleter)
             warnings.warn(
                 "Exporting an unversioned DLPack capsule is deprecated, "
                 "pass max_version=(1, 0) or higher.",
                 DeprecationWarning, stacklevel=2)
-            legacy_tensor = GetResultValue(ExportTensorToDLPack(self.sp_tensor))
-            return PyCapsule_New(legacy_tensor, 'dltensor', dlpack_pycapsule_deleter)
+            return capsule
 
         # Currently no major version other than legacy 0 and current 1.3
         dlm_tensor = GetResultValue(
diff --git a/python/pyarrow/tests/test_array.py b/python/pyarrow/tests/test_array.py
index 74ff837e16e..d9a14abe1a3 100644
--- a/python/pyarrow/tests/test_array.py
+++ b/python/pyarrow/tests/test_array.py
@@ -2737,6 +2737,10 @@ def test_array_from_list_of_timestamps(unit):
 
 
 @pytest.mark.numpy
+@pytest.mark.filterwarnings(
+    "ignore:The 'generic' unit for NumPy timedelta is deprecated:"
+    "DeprecationWarning"
+)
 def test_array_from_timestamp_with_generic_unit():
     if Version(np.__version__) >= Version("2.5.0"):
         pytest.skip("generic units of timedelta64 deprecated")
@@ -2779,6 +2783,10 @@ def test_array_from_numpy_timedelta(dtype, type):
 
 
 @pytest.mark.numpy
+@pytest.mark.filterwarnings(
+    "ignore:The 'generic' unit for NumPy timedelta is deprecated:"
+    "DeprecationWarning"
+)
 def test_array_from_numpy_timedelta_incorrect_unit():
     # generic (no unit)
     if Version(np.__version__) < Version("2.5.0"):
diff --git a/python/pyarrow/tests/test_dlpack.py b/python/pyarrow/tests/test_dlpack.py
index 0c3c082fd6e..88eab11e060 100644
--- a/python/pyarrow/tests/test_dlpack.py
+++ b/python/pyarrow/tests/test_dlpack.py
@@ -307,7 +307,7 @@ def test_dlpack_versioned_capsule(obj, max_version, copy):
 @check_bytes_allocated
 @pytest.mark.parametrize('obj', dlpack_objects())
 def test_dlpack_versioned_roundtrip(obj):
-    expected = np.from_dlpack(DLPackForwarder(obj, max_version=None))
+    expected = np.from_dlpack(DLPackForwarder(obj, max_version=(1, 0)))
     for copy in [None, False, True]:
         result = np.from_dlpack(
             DLPackForwarder(obj, max_version=(1, 0), copy=copy))
@@ -335,24 +335,24 @@ def test_dlpack_not_supported():
     arr = pa.array([1, None, 3])
     with pytest.raises(TypeError, match="Can only use DLPack "
                        "on arrays with no nulls."):
-        np.from_dlpack(arr)
+        np.from_dlpack(DLPackForwarder(arr, max_version=(1, 0)))
 
     arr = pa.array(
         [[0, 1], [3, 4]],
         type=pa.list_(pa.int32())
     )
     with pytest.raises(TypeError, match="DataType is not compatible with DLPack spec"):
-        np.from_dlpack(arr)
+        np.from_dlpack(DLPackForwarder(arr, max_version=(1, 0)))
 
     arr = pa.array([])
     with pytest.raises(TypeError, match="DataType is not compatible with DLPack spec"):
-        np.from_dlpack(arr)
+        np.from_dlpack(DLPackForwarder(arr, max_version=(1, 0)))
 
     # DLPack doesn't support bit-packed boolean values
     arr = pa.array([True, False, True])
     with pytest.raises(TypeError, match="Bit-packed boolean data type "
                        "not supported by DLPack."):
-        np.from_dlpack(arr)
+        np.from_dlpack(DLPackForwarder(arr, max_version=(1, 0)))
 
 
 def test_dlpack_cuda_not_supported():