diff --git a/doc/whats_new/v1.1.rst b/doc/whats_new/v1.1.rst
index 542636b1642f7..208c950c6e43d 100644
--- a/doc/whats_new/v1.1.rst
+++ b/doc/whats_new/v1.1.rst
@@ -162,6 +162,15 @@ Changelog
   ndarray with `np.nan` when passed a `Float32` or `Float64` pandas extension
   array with `pd.NA`. :pr:`21278` by `Thomas Fan`_.
 
+:mod:`sklearn.random_projection`
+................................
+
+- |API| Adds :term:`get_feature_names_out` to all transformers in the
+  :mod:`~sklearn.random_projection` module:
+  :class:`~sklearn.random_projection.GaussianRandomProjection` and
+  :class:`~sklearn.random_projection.SparseRandomProjection`. :pr:`21330` by
+  :user:`Loïc Estève <lesteve>`.
+
 Code and Documentation Contributors
 -----------------------------------
 
diff --git a/sklearn/random_projection.py b/sklearn/random_projection.py
index 6b2c9217713e0..3ddbecb677710 100644
--- a/sklearn/random_projection.py
+++ b/sklearn/random_projection.py
@@ -34,6 +34,7 @@
 import scipy.sparse as sp
 
 from .base import BaseEstimator, TransformerMixin
+from .base import _ClassNamePrefixFeaturesOutMixin
 
 from .utils import check_random_state
 from .utils.extmath import safe_sparse_dot
@@ -290,7 +291,9 @@ def _sparse_random_matrix(n_components, n_features, density="auto", random_state
         return np.sqrt(1 / density) / np.sqrt(n_components) * components
 
 
-class BaseRandomProjection(TransformerMixin, BaseEstimator, metaclass=ABCMeta):
+class BaseRandomProjection(
+    TransformerMixin, BaseEstimator, _ClassNamePrefixFeaturesOutMixin, metaclass=ABCMeta
+):
     """Base class for random projections.
 
     Warning: This class should not be used directly.
@@ -420,6 +423,14 @@ def transform(self, X):
         X_new = safe_sparse_dot(X, self.components_.T, dense_output=self.dense_output)
         return X_new
 
+    @property
+    def _n_features_out(self):
+        """Number of transformed output features.
+
+        Used by _ClassNamePrefixFeaturesOutMixin.get_feature_names_out.
+        """
+        return self.n_components
+
 
 class GaussianRandomProjection(BaseRandomProjection):
     """Reduce dimensionality through Gaussian random projection.
diff --git a/sklearn/tests/test_common.py b/sklearn/tests/test_common.py
index a476ba7dc8da5..c3eade24be412 100644
--- a/sklearn/tests/test_common.py
+++ b/sklearn/tests/test_common.py
@@ -371,7 +371,6 @@ def test_pandas_column_name_consistency(estimator):
     "manifold",
     "neighbors",
     "neural_network",
-    "random_projection",
 ]
 
 
diff --git a/sklearn/tests/test_random_projection.py b/sklearn/tests/test_random_projection.py
index 5866fde29d73b..1e894d906a3ad 100644
--- a/sklearn/tests/test_random_projection.py
+++ b/sklearn/tests/test_random_projection.py
@@ -24,7 +24,7 @@
 
 all_SparseRandomProjection: List[Any] = [SparseRandomProjection]
 all_DenseRandomProjection: List[Any] = [GaussianRandomProjection]
-all_RandomProjection = set(all_SparseRandomProjection + all_DenseRandomProjection)
+all_RandomProjection = all_SparseRandomProjection + all_DenseRandomProjection
 
 
 # Make some random data with uniformly located non zero entries with
@@ -359,3 +359,17 @@ def test_johnson_lindenstrauss_min_dim():
     Regression test for #17111: before #19374, 32-bit systems would fail.
     """
     assert johnson_lindenstrauss_min_dim(100, eps=1e-5) == 368416070986
+
+
+@pytest.mark.parametrize("random_projection_cls", all_RandomProjection)
+def test_random_projection_feature_names_out(random_projection_cls):
+    random_projection = random_projection_cls(n_components=2)
+    random_projection.fit(data)
+    names_out = random_projection.get_feature_names_out()
+    class_name_lower = random_projection_cls.__name__.lower()
+    expected_names_out = np.array(
+        [f"{class_name_lower}{i}" for i in range(random_projection.n_components_)],
+        dtype=object,
+    )
+
+    assert_array_equal(names_out, expected_names_out)