diff --git a/packages/eslint-plugin/docs/rules/promise-function-async.md b/packages/eslint-plugin/docs/rules/promise-function-async.md
index 5ccddf45ed47..79000974cead 100644
--- a/packages/eslint-plugin/docs/rules/promise-function-async.md
+++ b/packages/eslint-plugin/docs/rules/promise-function-async.md
@@ -6,7 +6,7 @@ Ensures that each function is only capable of:
 - returning a rejected promise, or
 - throwing an Error object.
 
-In contrast, non-`async` `Promise`-returning functions are technically capable of either.
+In contrast, non-`async` `Promise` - returning functions are technically capable of either.
 Code that handles the results of those functions will often need to handle both cases, which can get complex.
 This rule's practice removes a requirement for creating code to handle both cases.
 
@@ -17,18 +17,18 @@ Examples of **incorrect** code for this rule
 ```ts
 const arrowFunctionReturnsPromise = () => Promise.resolve('value');
 
-function functionDeturnsPromise() {
-  return Math.random() > 0.5 ? Promise.resolve('value') : false;
+function functionReturnsPromise() {
+  return Promise.resolve('value');
 }
 ```
 
 Examples of **correct** code for this rule
 
 ```ts
-const arrowFunctionReturnsPromise = async () => 'value';
+const arrowFunctionReturnsPromise = async () => Promise.resolve('value');
 
-async function functionDeturnsPromise() {
-  return Math.random() > 0.5 ? 'value' : false;
+async function functionReturnsPromise() {
+  return Promise.resolve('value');
 }
 ```
 
diff --git a/packages/eslint-plugin/src/rules/promise-function-async.ts b/packages/eslint-plugin/src/rules/promise-function-async.ts
index 1575cf02dfce..d907f3d45a55 100644
--- a/packages/eslint-plugin/src/rules/promise-function-async.ts
+++ b/packages/eslint-plugin/src/rules/promise-function-async.ts
@@ -98,7 +98,11 @@ export default util.createRule<Options, MessageIds>({
       const returnType = checker.getReturnTypeOfSignature(signatures[0]);
 
       if (
-        !util.containsTypeByName(returnType, allowAny!, allAllowedPromiseNames)
+        !util.containsAllTypesByName(
+          returnType,
+          allowAny!,
+          allAllowedPromiseNames,
+        )
       ) {
         return;
       }
diff --git a/packages/eslint-plugin/src/util/types.ts b/packages/eslint-plugin/src/util/types.ts
index 6f769d7f950a..72213da6ebbf 100644
--- a/packages/eslint-plugin/src/util/types.ts
+++ b/packages/eslint-plugin/src/util/types.ts
@@ -8,9 +8,9 @@ import ts from 'typescript';
 /**
  * @param type Type being checked by name.
  * @param allowedNames Symbol names checking on the type.
- * @returns Whether the type is, extends, or contains any of the allowed names.
+ * @returns Whether the type is, extends, or contains all of the allowed names.
  */
-export function containsTypeByName(
+export function containsAllTypesByName(
   type: ts.Type,
   allowAny: boolean,
   allowedNames: Set<string>,
@@ -31,13 +31,16 @@ export function containsTypeByName(
   }
 
   if (isUnionOrIntersectionType(type)) {
-    return type.types.some(t => containsTypeByName(t, allowAny, allowedNames));
+    return type.types.every(t =>
+      containsAllTypesByName(t, allowAny, allowedNames),
+    );
   }
 
   const bases = type.getBaseTypes();
   return (
     typeof bases !== 'undefined' &&
-    bases.some(t => containsTypeByName(t, allowAny, allowedNames))
+    bases.length > 0 &&
+    bases.every(t => containsAllTypesByName(t, allowAny, allowedNames))
   );
 }
 
diff --git a/packages/eslint-plugin/tests/rules/promise-function-async.test.ts b/packages/eslint-plugin/tests/rules/promise-function-async.test.ts
index 46100db7982a..66a65a21ffd3 100644
--- a/packages/eslint-plugin/tests/rules/promise-function-async.test.ts
+++ b/packages/eslint-plugin/tests/rules/promise-function-async.test.ts
@@ -106,6 +106,26 @@ function returnsUnknown(): unknown {
         },
       ],
     },
+    {
+      code: `
+interface ReadableStream {}
+interface Options {
+  stream: ReadableStream;
+}
+
+type Return = ReadableStream | Promise<void>;
+const foo = (options: Options): Return => {
+  return options.stream ? asStream(options) : asPromise(options);
+}
+      `,
+    },
+    {
+      code: `
+function foo(): Promise<string> | boolean {
+  return Math.random() > 0.5 ? Promise.resolve('value') : false;
+}
+      `,
+    },
   ],
   invalid: [
     {
@@ -379,5 +399,24 @@ const returnAllowedType = () => new PromiseType();
         },
       ],
     },
+    {
+      code: `
+interface SPromise<T> extends Promise<T> {}
+function foo(): Promise<string> | SPromise<boolean> {
+  return Math.random() > 0.5 ? Promise.resolve('value') : Promise.resolve(false);
+}
+      `,
+      options: [
+        {
+          allowedPromiseNames: ['SPromise'],
+        },
+      ],
+      errors: [
+        {
+          line: 3,
+          messageId,
+        },
+      ],
+    },
   ],
 });