diff --git a/doc/source/whatsnew/v0.24.0.txt b/doc/source/whatsnew/v0.24.0.txt
index b71edcf1f6f51..851c1a3fbd6e9 100644
--- a/doc/source/whatsnew/v0.24.0.txt
+++ b/doc/source/whatsnew/v0.24.0.txt
@@ -655,6 +655,7 @@ Datetimelike
 - Bug in :class:`DatetimeIndex` incorrectly allowing indexing with ``Timedelta`` object (:issue:`20464`)
 - Bug in :class:`DatetimeIndex` where frequency was being set if original frequency was ``None`` (:issue:`22150`)
 - Bug in rounding methods of :class:`DatetimeIndex` (:meth:`~DatetimeIndex.round`, :meth:`~DatetimeIndex.ceil`, :meth:`~DatetimeIndex.floor`) and :class:`Timestamp` (:meth:`~Timestamp.round`, :meth:`~Timestamp.ceil`, :meth:`~Timestamp.floor`) could give rise to loss of precision (:issue:`22591`)
+- Bug in :func:`to_datetime` with an :class:`Index` argument that would drop the ``name`` from the result (:issue:`21697`)
 
 Timedelta
 ^^^^^^^^^
diff --git a/pandas/core/tools/datetimes.py b/pandas/core/tools/datetimes.py
index 57387b9ea870a..4a5290a90313d 100644
--- a/pandas/core/tools/datetimes.py
+++ b/pandas/core/tools/datetimes.py
@@ -99,13 +99,13 @@ def _convert_and_box_cache(arg, cache_array, box, errors, name=None):
     result = Series(arg).map(cache_array)
     if box:
         if errors == 'ignore':
-            return Index(result)
+            return Index(result, name=name)
         else:
             return DatetimeIndex(result, name=name)
     return result.values
 
 
-def _return_parsed_timezone_results(result, timezones, box, tz):
+def _return_parsed_timezone_results(result, timezones, box, tz, name):
     """
     Return results from array_strptime if a %z or %Z directive was passed.
 
@@ -119,6 +119,9 @@ def _return_parsed_timezone_results(result, timezones, box, tz):
         True boxes result as an Index-like, False returns an ndarray
     tz : object
         None or pytz timezone object
+    name : string, default None
+        Name for a DatetimeIndex
+
     Returns
     -------
     tz_result : ndarray of parsed dates with timezone
@@ -136,7 +139,7 @@ def _return_parsed_timezone_results(result, timezones, box, tz):
                            in zip(result, timezones)])
     if box:
         from pandas import Index
-        return Index(tz_results)
+        return Index(tz_results, name=name)
     return tz_results
 
 
@@ -209,7 +212,7 @@ def _convert_listlike_datetimes(arg, box, format, name=None, tz=None,
         if box:
             if errors == 'ignore':
                 from pandas import Index
-                return Index(result)
+                return Index(result, name=name)
 
             return DatetimeIndex(result, tz=tz, name=name)
         return result
@@ -252,7 +255,7 @@ def _convert_listlike_datetimes(arg, box, format, name=None, tz=None,
                         arg, format, exact=exact, errors=errors)
                     if '%Z' in format or '%z' in format:
                         return _return_parsed_timezone_results(
-                            result, timezones, box, tz)
+                            result, timezones, box, tz, name)
                 except tslibs.OutOfBoundsDatetime:
                     if errors == 'raise':
                         raise
diff --git a/pandas/tests/indexes/datetimes/test_tools.py b/pandas/tests/indexes/datetimes/test_tools.py
index cc6db8f5854c8..3b7d6a709230b 100644
--- a/pandas/tests/indexes/datetimes/test_tools.py
+++ b/pandas/tests/indexes/datetimes/test_tools.py
@@ -233,6 +233,15 @@ def test_to_datetime_parse_timezone_malformed(self, offset):
         with pytest.raises(ValueError):
             pd.to_datetime([date], format=fmt)
 
+    def test_to_datetime_parse_timezone_keeps_name(self):
+        # GH 21697
+        fmt = '%Y-%m-%d %H:%M:%S %z'
+        arg = pd.Index(['2010-01-01 12:00:00 Z'], name='foo')
+        result = pd.to_datetime(arg, format=fmt)
+        expected = pd.DatetimeIndex(['2010-01-01 12:00:00'], tz='UTC',
+                                    name='foo')
+        tm.assert_index_equal(result, expected)
+
 
 class TestToDatetime(object):
     def test_to_datetime_pydatetime(self):
@@ -765,6 +774,14 @@ def test_unit_rounding(self, cache):
         expected = pd.Timestamp('2015-06-19 19:55:31.877000093')
         assert result == expected
 
+    @pytest.mark.parametrize('cache', [True, False])
+    def test_unit_ignore_keeps_name(self, cache):
+        # GH 21697
+        expected = pd.Index([15e9] * 2, name='name')
+        result = pd.to_datetime(expected, errors='ignore', box=True, unit='s',
+                                cache=cache)
+        tm.assert_index_equal(result, expected)
+
     @pytest.mark.parametrize('cache', [True, False])
     def test_dataframe(self, cache):