diff --git a/homeassistant/components/azure_storage/__init__.py b/homeassistant/components/azure_storage/__init__.py
index 00e419fd3c9..78d85dd6a59 100644
--- a/homeassistant/components/azure_storage/__init__.py
+++ b/homeassistant/components/azure_storage/__init__.py
@@ -2,8 +2,8 @@
 
 from aiohttp import ClientTimeout
 from azure.core.exceptions import (
+    AzureError,
     ClientAuthenticationError,
-    HttpResponseError,
     ResourceNotFoundError,
 )
 from azure.core.pipeline.transport._aiohttp import (
@@ -70,7 +70,7 @@ async def async_setup_entry(
             translation_key="invalid_auth",
             translation_placeholders={CONF_ACCOUNT_NAME: entry.data[CONF_ACCOUNT_NAME]},
         ) from err
-    except HttpResponseError as err:
+    except AzureError as err:
         raise ConfigEntryNotReady(
             translation_domain=DOMAIN,
             translation_key="cannot_connect",
diff --git a/homeassistant/components/azure_storage/backup.py b/homeassistant/components/azure_storage/backup.py
index 4a9254213dc..54fd069a11f 100644
--- a/homeassistant/components/azure_storage/backup.py
+++ b/homeassistant/components/azure_storage/backup.py
@@ -8,7 +8,7 @@ import json
 import logging
 from typing import Any, Concatenate
 
-from azure.core.exceptions import HttpResponseError
+from azure.core.exceptions import AzureError, HttpResponseError, ServiceRequestError
 from azure.storage.blob import BlobProperties
 
 from homeassistant.components.backup import (
@@ -80,6 +80,20 @@ def handle_backup_errors[_R, **P](
                 f"Error during backup operation in {func.__name__}:"
                 f" Status {err.status_code}, message: {err.message}"
             ) from err
+        except ServiceRequestError as err:
+            raise BackupAgentError(
+                f"Timeout during backup operation in {func.__name__}"
+            ) from err
+        except AzureError as err:
+            _LOGGER.debug(
+                "Error during backup in %s: %s",
+                func.__name__,
+                err,
+                exc_info=True,
+            )
+            raise BackupAgentError(
+                f"Error during backup operation in {func.__name__}: {err}"
+            ) from err
 
     return wrapper
 
diff --git a/tests/components/azure_storage/test_backup.py b/tests/components/azure_storage/test_backup.py
index 7c5912a4981..ebb491c2b7c 100644
--- a/tests/components/azure_storage/test_backup.py
+++ b/tests/components/azure_storage/test_backup.py
@@ -6,7 +6,7 @@ from collections.abc import AsyncGenerator
 from io import StringIO
 from unittest.mock import ANY, Mock, patch
 
-from azure.core.exceptions import HttpResponseError
+from azure.core.exceptions import AzureError, HttpResponseError, ServiceRequestError
 from azure.storage.blob import BlobProperties
 import pytest
 
@@ -276,14 +276,33 @@ async def test_agents_error_on_download_not_found(
     assert mock_client.download_blob.call_count == 0
 
 
+@pytest.mark.parametrize(
+    ("error", "message"),
+    [
+        (
+            HttpResponseError("http error"),
+            "Error during backup operation in async_delete_backup: Status None, message: http error",
+        ),
+        (
+            ServiceRequestError("timeout"),
+            "Timeout during backup operation in async_delete_backup",
+        ),
+        (
+            AzureError("generic error"),
+            "Error during backup operation in async_delete_backup: generic error",
+        ),
+    ],
+)
 async def test_error_during_delete(
     hass: HomeAssistant,
     hass_ws_client: WebSocketGenerator,
     mock_client: MagicMock,
     mock_config_entry: MockConfigEntry,
+    error: Exception,
+    message: str,
 ) -> None:
     """Test the error wrapper."""
-    mock_client.delete_blob.side_effect = HttpResponseError("Failed to delete backup")
+    mock_client.delete_blob.side_effect = error
 
     client = await hass_ws_client(hass)
 
@@ -297,12 +316,7 @@ async def test_error_during_delete(
 
     assert response["success"]
     assert response["result"] == {
-        "agent_errors": {
-            f"{DOMAIN}.{mock_config_entry.entry_id}": (
-                "Error during backup operation in async_delete_backup: "
-                "Status None, message: Failed to delete backup"
-            )
-        }
+        "agent_errors": {f"{DOMAIN}.{mock_config_entry.entry_id}": message}
     }