diff --git a/qiskit/visualization/dag_visualization.py b/qiskit/visualization/dag_visualization.py
index ad2fca6e9b..dae97b51b0 100644
--- a/qiskit/visualization/dag_visualization.py
+++ b/qiskit/visualization/dag_visualization.py
@@ -174,10 +174,13 @@ def dag_drawer(dag, scale=0.7, filename=None, style="color"):
                         label = register_bit_labels.get(
                             node.wire, f"q_{dag.find_bit(node.wire).index}"
                         )
-                    else:
+                    elif isinstance(node.wire, Clbit):
                         label = register_bit_labels.get(
                             node.wire, f"c_{dag.find_bit(node.wire).index}"
                         )
+                    else:
+                        label = str(node.wire.name)
+
                     n["label"] = label
                     n["color"] = "black"
                     n["style"] = "filled"
@@ -187,10 +190,12 @@ def dag_drawer(dag, scale=0.7, filename=None, style="color"):
                         label = register_bit_labels.get(
                             node.wire, f"q[{dag.find_bit(node.wire).index}]"
                         )
-                    else:
+                    elif isinstance(node.wire, Clbit):
                         label = register_bit_labels.get(
                             node.wire, f"c[{dag.find_bit(node.wire).index}]"
                         )
+                    else:
+                        label = str(node.wire.name)
                     n["label"] = label
                     n["color"] = "black"
                     n["style"] = "filled"
@@ -203,8 +208,10 @@ def dag_drawer(dag, scale=0.7, filename=None, style="color"):
             e = {}
             if isinstance(edge, Qubit):
                 label = register_bit_labels.get(edge, f"q_{dag.find_bit(edge).index}")
-            else:
+            elif isinstance(edge, Clbit):
                 label = register_bit_labels.get(edge, f"c_{dag.find_bit(edge).index}")
+            else:
+                label = str(edge.name)
             e["label"] = label
             return e
 
diff --git a/releasenotes/notes/fix-var-wires-4ebc40e0b19df253.yaml b/releasenotes/notes/fix-var-wires-4ebc40e0b19df253.yaml
new file mode 100644
index 0000000000..7cd1e74806
--- /dev/null
+++ b/releasenotes/notes/fix-var-wires-4ebc40e0b19df253.yaml
@@ -0,0 +1,8 @@
+---
+fixes:
+  - |
+    Fixed an issue with :func:`.dag_drawer` and :meth:`.DAGCircuit.draw`
+    when attempting to visualize a :class:`.DAGCircuit` instance that contained
+    :class:`.Var` wires. The visualizer would raise an exception trying to
+    do this which has been fixed so the expected visualization will be
+    generated.
diff --git a/test/python/visualization/test_dag_drawer.py b/test/python/visualization/test_dag_drawer.py
index 4b920390e8..d789b1e706 100644
--- a/test/python/visualization/test_dag_drawer.py
+++ b/test/python/visualization/test_dag_drawer.py
@@ -16,12 +16,14 @@ import os
 import tempfile
 import unittest
 
-from qiskit.circuit import QuantumRegister, ClassicalRegister, QuantumCircuit, Qubit, Clbit
+from qiskit.circuit import QuantumRegister, ClassicalRegister, QuantumCircuit, Qubit, Clbit, Store
 from qiskit.visualization import dag_drawer
 from qiskit.exceptions import InvalidFileError
 from qiskit.visualization import VisualizationError
 from qiskit.converters import circuit_to_dag, circuit_to_dagdependency
 from qiskit.utils import optionals as _optionals
+from qiskit.dagcircuit import DAGCircuit
+from qiskit.circuit.classical import expr, types
 from .visualization import path_to_diagram_reference, QiskitVisualizationTestCase
 
 
@@ -108,6 +110,17 @@ class TestDagDrawer(QiskitVisualizationTestCase):
             image = Image.open(tmp_path)
             self.assertImagesAreEqual(image, image_ref, 0.1)
 
+    @unittest.skipUnless(_optionals.HAS_GRAPHVIZ, "Graphviz not installed")
+    @unittest.skipUnless(_optionals.HAS_PIL, "PIL not installed")
+    def test_dag_drawer_with_var_wires(self):
+        """Test visualization works with var nodes."""
+        a = expr.Var.new("a", types.Bool())
+        dag = DAGCircuit()
+        dag.add_input_var(a)
+        dag.apply_operation_back(Store(a, a), (), ())
+        image = dag_drawer(dag)
+        self.assertIsNotNone(image)
+
 
 if __name__ == "__main__":
     unittest.main(verbosity=2)