diff --git a/releasenotes/notes/wait-random-exponential-min-2a4b7eed9f002436.yaml b/releasenotes/notes/wait-random-exponential-min-2a4b7eed9f002436.yaml
new file mode 100644
index 0000000..34efd1c
--- /dev/null
+++ b/releasenotes/notes/wait-random-exponential-min-2a4b7eed9f002436.yaml
@@ -0,0 +1,4 @@
+---
+fixes:
+  - |
+    Respects `min` arg for `wait_random_exponential`
diff --git a/tenacity/wait.py b/tenacity/wait.py
index 3addbb9..dc3c850 100644
--- a/tenacity/wait.py
+++ b/tenacity/wait.py
@@ -197,7 +197,7 @@ class wait_random_exponential(wait_exponential):
 
     def __call__(self, retry_state: "RetryCallState") -> float:
         high = super().__call__(retry_state=retry_state)
-        return random.uniform(0, high)
+        return random.uniform(self.min, high)
 
 
 class wait_exponential_jitter(wait_base):
diff --git a/tests/test_tenacity.py b/tests/test_tenacity.py
index ecc0312..b76fec2 100644
--- a/tests/test_tenacity.py
+++ b/tests/test_tenacity.py
@@ -472,9 +472,17 @@ def test_wait_random_exponential(self):
             self._assert_inclusive_range(fn(make_retry_state(8, 0)), 0, 60.0)
             self._assert_inclusive_range(fn(make_retry_state(9, 0)), 0, 60.0)
 
-        fn = tenacity.wait_random_exponential(10, 5)
+        # max wait
+        max_wait = 5
+        fn = tenacity.wait_random_exponential(10, max_wait)
         for _ in range(1000):
-            self._assert_inclusive_range(fn(make_retry_state(1, 0)), 0.00, 5.00)
+            self._assert_inclusive_range(fn(make_retry_state(1, 0)), 0.00, max_wait)
+
+        # min wait
+        min_wait = 5
+        fn = tenacity.wait_random_exponential(min=min_wait)
+        for _ in range(1000):
+            self._assert_inclusive_range(fn(make_retry_state(1, 0)), min_wait, 5)
 
         # Default arguments exist
         fn = tenacity.wait_random_exponential()