diff --git a/cpp/src/arrow/array/array_view_test.cc b/cpp/src/arrow/array/array_view_test.cc
index dfa8021dde1..ca1c82bfc90 100644
--- a/cpp/src/arrow/array/array_view_test.cc
+++ b/cpp/src/arrow/array/array_view_test.cc
@@ -27,6 +27,7 @@
 #include "arrow/extension_type.h"
 #include "arrow/result.h"
 #include "arrow/status.h"
+#include "arrow/testing/extension_type.h"
 #include "arrow/testing/gtest_util.h"
 #include "arrow/type.h"
 #include "arrow/util/endian.h"
@@ -480,6 +481,32 @@ TEST(TestArrayView, ExtensionType) {
   CheckView(expected, arr);
 }
 
+TEST(TestArrayView, ExtensionTypeNestedStorage) {
+  auto ty1 = list_extension_type();
+  const auto& ext_type = static_cast<const ExtensionType&>(*ty1);
+  auto data = ArrayFromJSON(ext_type.storage_type(), "[[0, -1, 42], [5, 6]]")->data();
+  data->type = ty1;
+  auto arr = ext_type.MakeArray(data);
+  ASSERT_OK(arr->ValidateFull());
+
+  CheckView(arr, arr);
+}
+
+TEST(TestArrayView, StructAsStructWithExtensionField) {
+  auto ty1 = list_extension_type();
+  const auto& ext_type = static_cast<const ExtensionType&>(*ty1);
+  auto ext_data = ArrayFromJSON(ext_type.storage_type(), "[[0, -1, 42], [5, 6]]")->data();
+  ext_data->type = ty1;
+  auto ext_arr = ext_type.MakeArray(ext_data);
+
+  auto sibling = ArrayFromJSON(int8(), "[1, 2]");
+  ASSERT_OK_AND_ASSIGN(auto arr, StructArray::Make({ext_arr, sibling},
+                                                   std::vector<std::string>{"a", "b"}));
+  ASSERT_OK(arr->ValidateFull());
+
+  CheckView(arr, arr);
+}
+
 TEST(TestArrayView, NonZeroOffset) {
   auto arr = ArrayFromJSON(int16(), "[10, 11, 12, 13]");
 
diff --git a/cpp/src/arrow/array/data.cc b/cpp/src/arrow/array/data.cc
index dcc659e7094..edeb1c31190 100644
--- a/cpp/src/arrow/array/data.cc
+++ b/cpp/src/arrow/array/data.cc
@@ -744,7 +744,12 @@ namespace {
 void AccumulateLayouts(const std::shared_ptr<DataType>& type,
                        std::vector<DataTypeLayout>* layouts) {
   layouts->push_back(type->layout());
-  for (const auto& child : type->fields()) {
+  const DataType* type_for_children = type.get();
+  if (type->id() == Type::EXTENSION) {
+    const auto& ext_type = checked_cast<const ExtensionType&>(*type);
+    type_for_children = ext_type.storage_type().get();
+  }
+  for (const auto& child : type_for_children->fields()) {
     AccumulateLayouts(child->type(), layouts);
   }
 }
@@ -916,8 +921,14 @@ struct ViewDataImpl {
         out_type, out_length, std::move(out_buffers), out_null_count, out_offset);
     out_data->dictionary = dictionary;
 
+    const DataType* type_for_children = out_type.get();
+    if (out_type->id() == Type::EXTENSION) {
+      const auto& ext_type = checked_cast<const ExtensionType&>(*out_type);
+      type_for_children = ext_type.storage_type().get();
+    }
+
     // Process children recursively, depth-first
-    for (const auto& child_field : out_type->fields()) {
+    for (const auto& child_field : type_for_children->fields()) {
       std::shared_ptr<ArrayData> child_data;
       RETURN_NOT_OK(MakeDataView(child_field, &child_data));
       out_data->child_data.push_back(std::move(child_data));
diff --git a/python/pyarrow/tests/test_extension_type.py b/python/pyarrow/tests/test_extension_type.py
index 35b801eca87..ce2870f795d 100644
--- a/python/pyarrow/tests/test_extension_type.py
+++ b/python/pyarrow/tests/test_extension_type.py
@@ -784,6 +784,38 @@ def test_cast_to_extension_with_nested_storage():
     assert result.equals(expected)
 
 
+def test_cast_table_with_extension_type_in_struct():
+    # https://github.com/apache/arrow/issues/37004
+
+    # fixed-size list
+    array = pa.array([[1, 2], [3, 4], [5, 6]], pa.list_(pa.int32(), 2))
+    ext_type = MyFixedListType(pa.list_(pa.int32(), 2))
+    ext_array = pa.ExtensionArray.from_storage(ext_type, array)
+    struct_array = pa.StructArray.from_arrays([ext_array], "x")
+
+    table = pa.Table.from_arrays([struct_array], ["field1"])
+    result = table.cast(table.schema)
+    assert result.schema == table.schema
+    assert result.equals(table)
+
+    result = struct_array.cast(struct_array.type)
+    assert result.equals(struct_array)
+
+    # variable-size list
+    array = pa.array([[1, 2], [3, 4, 5], [6]], pa.list_(pa.int32()))
+    ext_type = MyListType(pa.list_(pa.int32()))
+    ext_array = pa.ExtensionArray.from_storage(ext_type, array)
+    struct_array = pa.StructArray.from_arrays([ext_array], "x")
+
+    table = pa.Table.from_arrays([struct_array], ["field1"])
+    result = table.cast(table.schema)
+    assert result.schema == table.schema
+    assert result.equals(table)
+
+    result = struct_array.cast(struct_array.type)
+    assert result.equals(struct_array)
+
+
 def test_concat():
     arr1 = pa.array([1, 2, 3], IntegerType())
     arr2 = pa.array([4, 5, 6], IntegerType())