diff --git a/asv_bench/benchmarks/multiindex_object.py b/asv_bench/benchmarks/multiindex_object.py
index ca1d71f50bd5f..89d74e784b64c 100644
--- a/asv_bench/benchmarks/multiindex_object.py
+++ b/asv_bench/benchmarks/multiindex_object.py
@@ -8,6 +8,7 @@
     MultiIndex,
     RangeIndex,
     Series,
+    array,
     date_range,
 )
 
@@ -176,6 +177,20 @@ def time_sortlevel_one(self):
         self.mi.sortlevel(1)
 
 
+class SortValues:
+
+    params = ["int64", "Int64"]
+    param_names = ["dtype"]
+
+    def setup(self, dtype):
+        a = array(np.tile(np.arange(100), 1000), dtype=dtype)
+        b = array(np.tile(np.arange(1000), 100), dtype=dtype)
+        self.mi = MultiIndex.from_arrays([a, b])
+
+    def time_sort_values(self, dtype):
+        self.mi.sort_values()
+
+
 class Values:
     def setup_cache(self):
 
diff --git a/doc/source/whatsnew/v1.6.0.rst b/doc/source/whatsnew/v1.6.0.rst
index c393b8a57f805..59c27560c4e1e 100644
--- a/doc/source/whatsnew/v1.6.0.rst
+++ b/doc/source/whatsnew/v1.6.0.rst
@@ -101,6 +101,7 @@ Deprecations
 Performance improvements
 ~~~~~~~~~~~~~~~~~~~~~~~~
 - Performance improvement in :meth:`.GroupBy.median` for nullable dtypes (:issue:`37493`)
+- Performance improvement in :meth:`MultiIndex.argsort` and :meth:`MultiIndex.sort_values` (:issue:`48406`)
 - Performance improvement in :meth:`.GroupBy.mean` and :meth:`.GroupBy.var` for extension array dtypes (:issue:`37493`)
 - Performance improvement for :meth:`MultiIndex.unique` (:issue:`48335`)
 -
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 6bf00bc5bc366..0ebd1a3d718d8 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -2225,6 +2225,10 @@ def append(self, other):
             return Index._with_infer(new_tuples)
 
     def argsort(self, *args, **kwargs) -> npt.NDArray[np.intp]:
+        if len(args) == 0 and len(kwargs) == 0:
+            # np.lexsort is significantly faster than self._values.argsort()
+            values = [self._get_level_values(i) for i in reversed(range(self.nlevels))]
+            return np.lexsort(values)
         return self._values.argsort(*args, **kwargs)
 
     @Appender(_index_shared_docs["repeat"] % _index_doc_kwargs)
diff --git a/pandas/tests/indexes/multi/test_setops.py b/pandas/tests/indexes/multi/test_setops.py
index d29abc1bc6a9f..e15809a751b73 100644
--- a/pandas/tests/indexes/multi/test_setops.py
+++ b/pandas/tests/indexes/multi/test_setops.py
@@ -530,11 +530,6 @@ def test_union_duplicates(index, request):
     if index.empty or isinstance(index, (IntervalIndex, CategoricalIndex)):
         # No duplicates in empty indexes
         return
-    if index.dtype.kind == "c":
-        mark = pytest.mark.xfail(
-            reason="sort_values() call raises bc complex objects are not comparable"
-        )
-        request.node.add_marker(mark)
 
     values = index.unique().values.tolist()
     mi1 = MultiIndex.from_arrays([values, [1] * len(values)])