diff --git a/CHANGELOG.rst b/CHANGELOG.rst
index a41353c..a75559e 100644
--- a/CHANGELOG.rst
+++ b/CHANGELOG.rst
@@ -2,6 +2,10 @@
 Changelog
 =========
 
+* Fix ``SystemError`` on Python 3.13 and Windows when starting time travelling.
+
+  Thanks to Bernát Gábor for the report in `Issue #456 <https://github.com/adamchainz/time-machine/issues/456>`__.
+
 2.14.1 (2024-03-22)
 -------------------
 
diff --git a/src/_time_machine.c b/src/_time_machine.c
index 55e434b..9986c87 100644
--- a/src/_time_machine.c
+++ b/src/_time_machine.c
@@ -423,21 +423,25 @@ _time_machine_patch_if_needed(PyObject *module, PyObject *unused)
 
     PyObject *time_module = PyImport_ImportModule("time");
 
-
-
-    PyCFunctionObject *time_clock_gettime = (PyCFunctionObject *) PyObject_GetAttrString(time_module, "clock_gettime");
     /*
-        time.clock_gettime() is not always available
-        e.g. on builds against old macOS = official Python.org installer
+        time.clock_gettime(), only available on Unix platforms.
     */
-    if (time_clock_gettime != NULL) {
+    PyCFunctionObject *time_clock_gettime = (PyCFunctionObject *) PyObject_GetAttrString(time_module, "clock_gettime");
+    if (time_clock_gettime == NULL) {
+        PyErr_Clear();
+    } else {
         state->original_clock_gettime = time_clock_gettime->m_ml->ml_meth;
         time_clock_gettime->m_ml->ml_meth = _time_machine_clock_gettime;
         Py_DECREF(time_clock_gettime);
     }
 
+    /*
+        time.clock_gettime_ns(), only available on Unix platforms.
+    */
     PyCFunctionObject *time_clock_gettime_ns = (PyCFunctionObject *) PyObject_GetAttrString(time_module, "clock_gettime_ns");
-    if (time_clock_gettime_ns != NULL) {
+    if (time_clock_gettime_ns == NULL) {
+        PyErr_Clear();
+    } else {
         state->original_clock_gettime_ns = time_clock_gettime_ns->m_ml->ml_meth;
         time_clock_gettime_ns->m_ml->ml_meth = _time_machine_clock_gettime_ns;
         Py_DECREF(time_clock_gettime_ns);