diff --git a/python/pyarrow/array.pxi b/python/pyarrow/array.pxi
index 3060c533255..ce248f4793e 100644
--- a/python/pyarrow/array.pxi
+++ b/python/pyarrow/array.pxi
@@ -1123,6 +1123,19 @@ cdef PandasOptions _convert_pandas_options(dict options):
     return result
 
 
+def _compute_binary_op(func_name, left, right):
+    """
+    Helper for arithmetic/bitwise dunder methods.
+
+    Only use for ops that can't raise ArrowTypeError as it
+    subclasses TypeError, so will get swallowed.
+    """
+    try:
+        return _pc().call_function(func_name, [left, right])
+    except TypeError:
+        return NotImplemented
+
+
 cdef class Array(_PandasConvertible):
     """
     The base class for all Arrow arrays.
@@ -2440,15 +2453,15 @@ cdef class Array(_PandasConvertible):
 
     def __add__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('add_checked', [self, other])
+        return _compute_binary_op('add_checked', self, other)
 
     def __truediv__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('divide_checked', [self, other])
+        return _compute_binary_op('divide_checked', self, other)
 
     def __mul__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('multiply_checked', [self, other])
+        return _compute_binary_op('multiply_checked', self, other)
 
     def __neg__(self):
         self._assert_cpu()
@@ -2456,31 +2469,31 @@ cdef class Array(_PandasConvertible):
 
     def __pow__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('power_checked', [self, other])
+        return _compute_binary_op('power_checked', self, other)
 
     def __sub__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('subtract_checked', [self, other])
+        return _compute_binary_op('subtract_checked', self, other)
 
     def __and__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('bit_wise_and', [self, other])
+        return _compute_binary_op('bit_wise_and', self, other)
 
     def __or__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('bit_wise_or', [self, other])
+        return _compute_binary_op('bit_wise_or', self, other)
 
     def __xor__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('bit_wise_xor', [self, other])
+        return _compute_binary_op('bit_wise_xor', self, other)
 
     def __lshift__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('shift_left_checked', [self, other])
+        return _compute_binary_op('shift_left_checked', self, other)
 
     def __rshift__(self, object other):
         self._assert_cpu()
-        return _pc().call_function('shift_right_checked', [self, other])
+        return _compute_binary_op('shift_right_checked', self, other)
 
 
 cdef _array_like_to_pandas(obj, options, types_mapper):
diff --git a/python/pyarrow/scalar.pxi b/python/pyarrow/scalar.pxi
index 863ab2b66a8..f96fb40572b 100644
--- a/python/pyarrow/scalar.pxi
+++ b/python/pyarrow/scalar.pxi
@@ -199,37 +199,37 @@ cdef class Scalar(_Weakrefable):
         return _pc().call_function('abs_checked', [self])
 
     def __add__(self, object other):
-        return _pc().call_function('add_checked', [self, other])
+        return _compute_binary_op('add_checked', self, other)
 
     def __truediv__(self, object other):
-        return _pc().call_function('divide_checked', [self, other])
+        return _compute_binary_op('divide_checked', self, other)
 
     def __mul__(self, object other):
-        return _pc().call_function('multiply_checked', [self, other])
+        return _compute_binary_op('multiply_checked', self, other)
 
     def __neg__(self):
         return _pc().call_function('negate_checked', [self])
 
     def __pow__(self, object other):
-        return _pc().call_function('power_checked', [self, other])
+        return _compute_binary_op('power_checked', self, other)
 
     def __sub__(self, object other):
-        return _pc().call_function('subtract_checked', [self, other])
+        return _compute_binary_op('subtract_checked', self, other)
 
     def __and__(self, object other):
-        return _pc().call_function('bit_wise_and', [self, other])
+        return _compute_binary_op('bit_wise_and', self, other)
 
     def __or__(self, object other):
-        return _pc().call_function('bit_wise_or', [self, other])
+        return _compute_binary_op('bit_wise_or', self, other)
 
     def __xor__(self, object other):
-        return _pc().call_function('bit_wise_xor', [self, other])
+        return _compute_binary_op('bit_wise_xor', self, other)
 
     def __lshift__(self, object other):
-        return _pc().call_function('shift_left_checked', [self, other])
+        return _compute_binary_op('shift_left_checked', self, other)
 
     def __rshift__(self, object other):
-        return _pc().call_function('shift_right_checked', [self, other])
+        return _compute_binary_op('shift_right_checked', self, other)
 
 
 _NULL = NA = None
diff --git a/python/pyarrow/tests/test_array.py b/python/pyarrow/tests/test_array.py
index a1e3616c9ce..9d3c6014a81 100644
--- a/python/pyarrow/tests/test_array.py
+++ b/python/pyarrow/tests/test_array.py
@@ -21,6 +21,7 @@
 import hypothesis as h
 import hypothesis.strategies as st
 import itertools
+import operator
 import pytest
 import struct
 import subprocess
@@ -4612,3 +4613,36 @@ def test_dictionary_uint64_index_to_pandas():
     result = arr.to_pandas()
     assert list(result.cat.categories) == ["a", "b"]
     assert result.cat.codes.tolist() == [0, 1, -1, 0]
+
+
+@pytest.mark.parametrize("op", [
+    operator.add,
+    operator.sub,
+    operator.mul,
+    operator.truediv,
+    operator.pow,
+    operator.and_,
+    operator.or_,
+    operator.xor,
+    operator.lshift,
+    operator.rshift,
+])
+def test_arithmetic_dunders_unknown_types(op):
+    # GH-49826
+    class MyObj:
+        def __radd__(self, other):
+            return "reflected"
+
+        __rsub__ = __rmul__ = __rtruediv__ = __rpow__ = __radd__
+        __rand__ = __ror__ = __rxor__ = __rlshift__ = __rrshift__ = __radd__
+
+    assert op(pa.array([1, 2, 3]), MyObj()) == "reflected"
+
+    with pytest.raises(TypeError, match="unsupported operand type"):
+        op(pa.array([1, 2, 3]), object())
+
+
+def test_arithmetic_dunder_raises_arrow_invalid():
+    # GH-49826
+    with pytest.raises(pa.ArrowInvalid, match="divide by zero"):
+        pa.array([1, 2, 3]) / pa.scalar(0)
diff --git a/python/pyarrow/tests/test_scalars.py b/python/pyarrow/tests/test_scalars.py
index 08f9fcd55ce..a71915b73aa 100644
--- a/python/pyarrow/tests/test_scalars.py
+++ b/python/pyarrow/tests/test_scalars.py
@@ -17,6 +17,7 @@
 
 import datetime
 import decimal
+import operator
 import pytest
 import weakref
 from collections.abc import Sequence, Mapping
@@ -1051,3 +1052,36 @@ def test_dunders_checked_overflow():
         scl ** scl
     with pytest.raises(pa.ArrowInvalid, match=error_match):
         scl * scl
+
+
+@pytest.mark.parametrize("op", [
+    operator.add,
+    operator.sub,
+    operator.mul,
+    operator.truediv,
+    operator.pow,
+    operator.and_,
+    operator.or_,
+    operator.xor,
+    operator.lshift,
+    operator.rshift,
+])
+def test_arithmetic_dunders_unknown_types(op):
+    # GH-49826
+    class MyObj:
+        def __radd__(self, other):
+            return "reflected"
+
+        __rsub__ = __rmul__ = __rtruediv__ = __rpow__ = __radd__
+        __rand__ = __ror__ = __rxor__ = __rlshift__ = __rrshift__ = __radd__
+
+    assert op(pa.scalar(5), MyObj()) == "reflected"
+
+    with pytest.raises(TypeError, match="unsupported operand type"):
+        op(pa.scalar(1), object())
+
+
+def test_arithmetic_dunder_raises_arrow_invalid():
+    # GH-49826
+    with pytest.raises(pa.ArrowInvalid, match="divide by zero"):
+        pa.scalar(1) / pa.scalar(0)