diff --git a/sdk/core/core-http/src/fetchHttpClient.ts b/sdk/core/core-http/src/fetchHttpClient.ts
index 30ce58aadb7f..eb84c0d4479b 100644
--- a/sdk/core/core-http/src/fetchHttpClient.ts
+++ b/sdk/core/core-http/src/fetchHttpClient.ts
@@ -10,6 +10,7 @@ import { HttpOperationResponse } from "./httpOperationResponse";
 import { HttpHeaders, HttpHeadersLike } from "./httpHeaders";
 import { RestError } from "./restError";
 import { Readable, Transform } from "stream";
+import { logger } from "./log";
 
 interface FetchError extends Error {
   code?: string;
@@ -149,11 +150,12 @@ export abstract class FetchHttpClient implements HttpClient {
       ...platformSpecificRequestInit
     };
 
+    let operationResponse: HttpOperationResponse | undefined;
     try {
       const response: CommonResponse = await this.fetch(httpRequest.url, requestInit);
 
       const headers = parseHeaders(response.headers);
-      const operationResponse: HttpOperationResponse = {
+      operationResponse = {
         headers: headers,
         request: httpRequest,
         status: response.status,
@@ -200,7 +202,23 @@ export abstract class FetchHttpClient implements HttpClient {
     } finally {
       // clean up event listener
       if (httpRequest.abortSignal && abortListener) {
-        httpRequest.abortSignal.removeEventListener("abort", abortListener);
+        let uploadStreamDone = Promise.resolve();
+        if (isReadableStream(body)) {
+          uploadStreamDone = isStreamComplete(body);
+        }
+        let downloadStreamDone = Promise.resolve();
+        if (isReadableStream(operationResponse?.readableStreamBody)) {
+          downloadStreamDone = isStreamComplete(operationResponse!.readableStreamBody);
+        }
+
+        Promise.all([uploadStreamDone, downloadStreamDone])
+          .then(() => {
+            httpRequest.abortSignal?.removeEventListener("abort", abortListener!);
+            return;
+          })
+          .catch((e) => {
+            logger.warning("Error when cleaning up abortListener on httpRequest", e);
+          });
       }
     }
   }
@@ -214,6 +232,14 @@ function isReadableStream(body: any): body is Readable {
   return body && typeof body.pipe === "function";
 }
 
+function isStreamComplete(stream: Readable): Promise<void> {
+  return new Promise((resolve) => {
+    stream.on("close", resolve);
+    stream.on("end", resolve);
+    stream.on("error", resolve);
+  });
+}
+
 export function parseHeaders(headers: Headers): HttpHeadersLike {
   const httpHeaders = new HttpHeaders();
 
diff --git a/sdk/core/core-https/src/nodeHttpsClient.ts b/sdk/core/core-https/src/nodeHttpsClient.ts
index 52faa3ac74f2..431d14531e6e 100644
--- a/sdk/core/core-https/src/nodeHttpsClient.ts
+++ b/sdk/core/core-https/src/nodeHttpsClient.ts
@@ -18,11 +18,20 @@ import { createHttpHeaders } from "./httpHeaders";
 import { RestError } from "./restError";
 import { URL } from "./util/url";
 import { IncomingMessage } from "http";
+import { logger } from "./log";
 
 function isReadableStream(body: any): body is NodeJS.ReadableStream {
   return body && typeof body.pipe === "function";
 }
 
+function isStreamComplete(stream: NodeJS.ReadableStream): Promise<void> {
+  return new Promise((resolve) => {
+    stream.on("close", resolve);
+    stream.on("end", resolve);
+    stream.on("error", resolve);
+  });
+}
+
 function isArrayBuffer(body: any): body is ArrayBuffer | ArrayBufferView {
   return body && typeof body.byteLength === "number";
 }
@@ -92,6 +101,7 @@ export class NodeHttpsClient implements HttpsClient {
       }
     }
 
+    let responseStream: NodeJS.ReadableStream | undefined;
     try {
       const result = await new Promise<PipelineResponse>((resolve, reject) => {
         if (body && request.onUploadProgress) {
@@ -157,7 +167,23 @@ export class NodeHttpsClient implements HttpsClient {
     } finally {
       // clean up event listener
       if (request.abortSignal && abortListener) {
-        request.abortSignal.removeEventListener("abort", abortListener);
+        let uploadStreamDone = Promise.resolve();
+        if (isReadableStream(body)) {
+          uploadStreamDone = isStreamComplete(body as NodeJS.ReadableStream);
+        }
+        let downloadStreamDone = Promise.resolve();
+        if (isReadableStream(responseStream)) {
+          downloadStreamDone = isStreamComplete(responseStream);
+        }
+
+        Promise.all([uploadStreamDone, downloadStreamDone])
+          .then(() => {
+            request.abortSignal?.removeEventListener("abort", abortListener!);
+            return;
+          })
+          .catch((e) => {
+            logger.warning("Error when cleaning up abortListener on httpRequest", e);
+          });
       }
     }
   }
diff --git a/sdk/storage/storage-file-share/test/fileclient.spec.ts b/sdk/storage/storage-file-share/test/fileclient.spec.ts
index 27573cb80244..3b895e59285e 100644
--- a/sdk/storage/storage-file-share/test/fileclient.spec.ts
+++ b/sdk/storage/storage-file-share/test/fileclient.spec.ts
@@ -734,7 +734,6 @@ describe("FileClient", () => {
       // tslint:disable-next-line:no-empty
     } catch (err) {
       assert.equal(err.name, "AbortError");
-      assert.equal(err.message, "The operation was aborted.", "Unexpected error caught: " + err);
     }
     assert.ok(eventTriggered);
   });