diff --git a/qiskit/extensions/quantum_initializer/squ.py b/qiskit/extensions/quantum_initializer/squ.py
index 0136898009f3..eb1c248e39de 100644
--- a/qiskit/extensions/quantum_initializer/squ.py
+++ b/qiskit/extensions/quantum_initializer/squ.py
@@ -59,7 +59,7 @@ def __init__(self, unitary_matrix, mode="ZYZ", up_to_diagonal=False, u=None):
         self._diag = None
 
         # Create new gate
-        super().__init__("unitary", 1, [unitary_matrix])
+        super().__init__("squ", 1, [unitary_matrix])
 
     def inverse(self):
         """Return the inverse.
diff --git a/qiskit/transpiler/preset_passmanagers/level0.py b/qiskit/transpiler/preset_passmanagers/level0.py
index e34cb4cd92a2..c306247022f1 100644
--- a/qiskit/transpiler/preset_passmanagers/level0.py
+++ b/qiskit/transpiler/preset_passmanagers/level0.py
@@ -105,7 +105,11 @@ def _choose_layout_condition(property_set):
     _embed = [FullAncillaAllocation(coupling_map), EnlargeWithAncilla(), ApplyLayout()]
 
     # 3. Decompose so only 1-qubit and 2-qubit gates remain
-    _unroll3q = Unroll3qOrMore()
+    _unroll3q = [
+        # Use unitary synthesis for basis aware decomposition of UnitaryGates
+        UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
+        Unroll3qOrMore(),
+    ]
 
     # 4. Swap to fit the coupling map
     _swap_check = CheckMap(coupling_map)
@@ -139,9 +143,14 @@ def _swap_condition(property_set):
     elif translation_method == "translator":
         from qiskit.circuit.equivalence_library import SessionEquivalenceLibrary as sel
 
-        _unroll = [UnrollCustomDefinitions(sel, basis_gates), BasisTranslator(sel, basis_gates)]
+        _unroll = [
+            UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
+            UnrollCustomDefinitions(sel, basis_gates),
+            BasisTranslator(sel, basis_gates),
+        ]
     elif translation_method == "synthesis":
         _unroll = [
+            UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
             Unroll3qOrMore(),
             Collect2qBlocks(),
             ConsolidateBlocks(basis_gates=basis_gates),
diff --git a/qiskit/transpiler/preset_passmanagers/level1.py b/qiskit/transpiler/preset_passmanagers/level1.py
index deab87f30467..05ea8c1225bc 100644
--- a/qiskit/transpiler/preset_passmanagers/level1.py
+++ b/qiskit/transpiler/preset_passmanagers/level1.py
@@ -125,7 +125,11 @@ def _not_perfect_yet(property_set):
     _embed = [FullAncillaAllocation(coupling_map), EnlargeWithAncilla(), ApplyLayout()]
 
     # 4. Decompose so only 1-qubit and 2-qubit gates remain
-    _unroll3q = Unroll3qOrMore()
+    _unroll3q = [
+        # Use unitary synthesis for basis aware decomposition of UnitaryGates
+        UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
+        Unroll3qOrMore(),
+    ]
 
     # 5. Swap to fit the coupling map
     _swap_check = CheckMap(coupling_map)
@@ -159,9 +163,18 @@ def _swap_condition(property_set):
     elif translation_method == "translator":
         from qiskit.circuit.equivalence_library import SessionEquivalenceLibrary as sel
 
-        _unroll = [UnrollCustomDefinitions(sel, basis_gates), BasisTranslator(sel, basis_gates)]
+        _unroll = [
+            # Use unitary synthesis for basis aware decomposition of UnitaryGates before
+            # custom unrolling
+            UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
+            UnrollCustomDefinitions(sel, basis_gates),
+            BasisTranslator(sel, basis_gates),
+        ]
     elif translation_method == "synthesis":
         _unroll = [
+            # Use unitary synthesis for basis aware decomposition of UnitaryGates before
+            # collection
+            UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
             Unroll3qOrMore(),
             Collect2qBlocks(),
             ConsolidateBlocks(basis_gates=basis_gates),
diff --git a/qiskit/transpiler/preset_passmanagers/level2.py b/qiskit/transpiler/preset_passmanagers/level2.py
index d26222b3cc61..6778e49d54bb 100644
--- a/qiskit/transpiler/preset_passmanagers/level2.py
+++ b/qiskit/transpiler/preset_passmanagers/level2.py
@@ -159,7 +159,11 @@ def _csp_not_found_match(property_set):
     _embed = [FullAncillaAllocation(coupling_map), EnlargeWithAncilla(), ApplyLayout()]
 
     # 3. Unroll to 1q or 2q gates
-    _unroll3q = Unroll3qOrMore()
+    _unroll3q = [
+        # Use unitary synthesis for basis aware decomposition of UnitaryGates
+        UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
+        Unroll3qOrMore(),
+    ]
 
     # 4. Swap to fit the coupling map
     _swap_check = CheckMap(coupling_map)
@@ -193,9 +197,18 @@ def _swap_condition(property_set):
     elif translation_method == "translator":
         from qiskit.circuit.equivalence_library import SessionEquivalenceLibrary as sel
 
-        _unroll = [UnrollCustomDefinitions(sel, basis_gates), BasisTranslator(sel, basis_gates)]
+        _unroll = [
+            # Use unitary synthesis for basis aware decomposition of UnitaryGates before
+            # custom unrolling
+            UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
+            UnrollCustomDefinitions(sel, basis_gates),
+            BasisTranslator(sel, basis_gates),
+        ]
     elif translation_method == "synthesis":
         _unroll = [
+            # Use unitary synthesis for basis aware decomposition of UnitaryGates before
+            # collection
+            UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
             Unroll3qOrMore(),
             Collect2qBlocks(),
             ConsolidateBlocks(basis_gates=basis_gates),
diff --git a/qiskit/transpiler/preset_passmanagers/level3.py b/qiskit/transpiler/preset_passmanagers/level3.py
index 257bc4babe88..ffe12c5918f4 100644
--- a/qiskit/transpiler/preset_passmanagers/level3.py
+++ b/qiskit/transpiler/preset_passmanagers/level3.py
@@ -100,7 +100,11 @@ def level_3_pass_manager(pass_manager_config: PassManagerConfig) -> PassManager:
     approximation_degree = pass_manager_config.approximation_degree
 
     # 1. Unroll to 1q or 2q gates
-    _unroll3q = Unroll3qOrMore()
+    _unroll3q = [
+        # Use unitary synthesis for basis aware decomposition of UnitaryGates
+        UnitarySynthesis(basis_gates, approximation_degree=approximation_degree),
+        Unroll3qOrMore(),
+    ]
 
     # 2. Layout on good qubits if calibration info available, otherwise on dense links
     _given_layout = SetLayout(initial_layout)
diff --git a/releasenotes/notes/squ-gate-name-785b7896300a92ef.yaml b/releasenotes/notes/squ-gate-name-785b7896300a92ef.yaml
new file mode 100644
index 000000000000..36608e046781
--- /dev/null
+++ b/releasenotes/notes/squ-gate-name-785b7896300a92ef.yaml
@@ -0,0 +1,9 @@
+---
+upgrade:
+  - |
+    The :attr:`~qiskit.extensions.SingleQubitUnitary.name` attribute of the
+    :class:`~qiskit.extensions.SingleQubitUnitary` gate class has been changed
+    from ``unitary`` to ``squ``. This was necessary to avoid a conflict with
+    the :class:`~qiskit.extensions.UnitaryGate` class's name which was also
+    ``unitary`` since the 2 gates are not the same and don't have the same
+    implementation (and can't be used interchangeably).