diff --git a/poetry.lock b/poetry.lock
index 5a970fc6f..f5543103b 100644
--- a/poetry.lock
+++ b/poetry.lock
@@ -469,7 +469,7 @@ toml = ["tomli"]
 name = "cryptography"
 version = "42.0.8"
 description = "cryptography is a package which provides cryptographic recipes and primitives to Python developers."
-optional = false
+optional = true
 python-versions = ">=3.7"
 files = [
     {file = "cryptography-42.0.8-cp37-abi3-macosx_10_12_universal2.whl", hash = "sha256:81d8a521705787afe7a18d5bfb47ea9d9cc068206270aad0b96a725022e18d2e"},
@@ -1528,7 +1528,7 @@ windows-terminal = ["colorama (>=0.4.6)"]
 name = "pyjwt"
 version = "2.8.0"
 description = "JSON Web Token implementation in Python"
-optional = false
+optional = true
 python-versions = ">=3.7"
 files = [
     {file = "PyJWT-2.8.0-py3-none-any.whl", hash = "sha256:59127c392cc44c2da5bb3192169a91f429924e17aff6534d70fdc02ab3e04320"},
@@ -2666,7 +2666,7 @@ test = ["big-O", "importlib-resources", "jaraco.functools", "jaraco.itertools",
 [extras]
 docs = ["furo", "myst-parser", "pytest", "sphinx", "sphinx-autobuild", "sphinx-copybutton", "sphinx-inline-tabs", "sphinx-notfound-page", "sphinx-reredirects"]
 faker = ["faker"]
-jwt = []
+jwt = ["PyJWT", "cryptography"]
 parquet = ["numpy", "numpy", "pyarrow"]
 s3 = ["fs-s3fs"]
 testing = ["pytest", "pytest-durations"]
@@ -2674,4 +2674,4 @@ testing = ["pytest", "pytest-durations"]
 [metadata]
 lock-version = "2.0"
 python-versions = ">=3.8"
-content-hash = "1316397863d23f10b9f8ec4b5647d4b95311ef90e98618dc3833cacf9ba1ef64"
+content-hash = "82c4b9443a3fed513d597831da8a953b3b2989e4859895939a31db30959a19d9"
diff --git a/pyproject.toml b/pyproject.toml
index 8734abe9a..990fb97f0 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -42,7 +42,6 @@ python = ">=3.8"
 backoff = { version = ">=2.0.0", python = "<4" }
 backports-datetime-fromisoformat = { version = ">=2.0.1", python = "<3.11" }
 click = "~=8.0"
-cryptography = ">=3.4.6"
 fs = ">=2.4.16"
 importlib-metadata = {version = "<9.0.0", python = "<3.12"}
 importlib-resources = {version = ">=5.12.0,!=6.2.0,!=6.3.0,!=6.3.1", python = "<3.9"}
@@ -52,7 +51,6 @@ jsonpath-ng = ">=1.5.3"
 jsonschema = ">=4.16.0"
 packaging = ">=23.1"
 pendulum = ">=2.1.0,<4"
-PyJWT = "~=2.4"
 python-dateutil = ">=2.8.2"
 python-dotenv = ">=0.20"
 PyYAML = ">=6.0"
@@ -93,10 +91,15 @@ pytest-durations = {version = ">=1.2.0", optional = true}
 # installed as optional 'faker' extra
 faker = {version = ">=22.5,<27.0", optional = true}
 
+# Crypto extras
+cryptography = { version = ">=3.4.6", optional = true }
+PyJWT = { version = "~=2.4", optional = true }
+
 [tool.poetry.extras]
-# TODO: Add 'cryptography' and 'PyJWT' to the 'jwt' when we want to remove them
-# from the main dependencies
-jwt = []
+jwt = [
+    "cryptography",
+    "PyJWT",
+]
 docs = [
     "furo",
     "myst-parser",
@@ -273,9 +276,9 @@ warn_unused_ignores = true
 ignore_missing_imports = true
 module = [
     "backports.datetime_fromisoformat.*",
-    "joblib.*",  # TODO: Remove when https://github.com/joblib/joblib/issues/1516 is shipped
-    "jsonpath_ng.*",
-    "pyarrow.*", # TODO: Remove when https://github.com/apache/arrow/issues/32609 if implemented and released
+    "joblib.*",       # TODO: Remove when https://github.com/joblib/joblib/issues/1516 is shipped
+    "jsonpath_ng.*",  # TODO: Remove when https://github.com/h2non/jsonpath-ng/issues/152 is implemented and released
+    "pyarrow.*",      # TODO: Remove when https://github.com/apache/arrow/issues/32609 if implemented and released
 ]
 
 [tool.poetry-dynamic-versioning]
diff --git a/singer_sdk/authenticators.py b/singer_sdk/authenticators.py
index 09f42efee..21ef1ba22 100644
--- a/singer_sdk/authenticators.py
+++ b/singer_sdk/authenticators.py
@@ -568,13 +568,18 @@ def oauth_request_payload(self) -> dict:
             Payload object for OAuth.
 
         Raises:
+            RuntimeError: If the JWT dependencies are not installed.
             ValueError: If the private key is not set.
         """
-        import jwt  # noqa: PLC0415
-        from cryptography.hazmat.backends import default_backend  # noqa: PLC0415
-        from cryptography.hazmat.primitives import serialization  # noqa: PLC0415
+        try:
+            import jwt  # noqa: PLC0415
+            from cryptography.hazmat.backends import default_backend  # noqa: PLC0415
+            from cryptography.hazmat.primitives import serialization  # noqa: PLC0415
+        except ModuleNotFoundError as ex:  # pragma: no cover
+            msg = "Install singer-sdk[jwt] to use OAuthJWTAuthenticator."
+            raise RuntimeError(msg) from ex
 
-        if not self.private_key:
+        if not self.private_key:  # pragma: no cover
             msg = "Missing 'private_key' property for OAuth payload."
             raise ValueError(msg)