diff --git a/src/util/StreamUtil.ts b/src/util/StreamUtil.ts
index 555c27825..c878fc0f7 100644
--- a/src/util/StreamUtil.ts
+++ b/src/util/StreamUtil.ts
@@ -40,6 +40,16 @@ export function pipeSafely<T extends Writable>(readable: NodeJS.ReadableStream,
     // in order to prevent memory leaks."
     destination.destroy(mapError ? mapError(error) : error);
   });
+
+  // Make sure we have no dangling streams in case of unpiping, which can happen if there's an error.
+  // This can also happen if a stream naturally closes so most calls here will not be indication of a problem.
+  // From https://nodejs.org/api/stream.html#stream_errors_while_writing :
+  // "If a Readable stream pipes into a Writable stream when Writable emits an error,
+  // the Readable stream will be unpiped."
+  destination.on('unpipe', (source): void => {
+    source.destroy();
+  });
+
   return guardStream(destination);
 }
 
diff --git a/test/unit/util/StreamUtil.test.ts b/test/unit/util/StreamUtil.test.ts
index a9ce2dc66..70bfeb448 100644
--- a/test/unit/util/StreamUtil.test.ts
+++ b/test/unit/util/StreamUtil.test.ts
@@ -40,6 +40,22 @@ describe('StreamUtil', (): void => {
       const piped = pipeSafely(input, output, (): any => new Error('other error'));
       await expect(readableToString(piped)).rejects.toThrow('other error');
     });
+
+    it('destroys the source stream in case the destinations becomes unpiped.', async(): Promise<void> => {
+      const input = new PassThrough();
+      const output = new PassThrough();
+      const piped = pipeSafely(input, output);
+
+      // Catch errors to prevent problems in test output
+      output.on('error', (): void => {
+        // Empty
+      });
+
+      piped.destroy(new Error('this causes an unpipe!'));
+      // Allow events to propagate
+      await new Promise(setImmediate);
+      expect(input.destroyed).toBe(true);
+    });
   });
 
   describe('#transformSafely', (): void => {