diff --git a/src/wp-includes/pluggable.php b/src/wp-includes/pluggable.php
index 511e6acac6..5c7747774b 100644
--- a/src/wp-includes/pluggable.php
+++ b/src/wp-includes/pluggable.php
@@ -349,7 +349,7 @@ function wp_mail( $to, $subject, $message, $headers = '', $attachments = array()
 	 */
 	$from_name = apply_filters( 'wp_mail_from_name', $from_name );
 
-	$phpmailer->setFrom( $from_email, $from_name );
+	$phpmailer->setFrom( $from_email, $from_name, false );
 
 	// Set destination addresses
 	if ( !is_array( $to ) )
diff --git a/tests/phpunit/tests/mail.php b/tests/phpunit/tests/mail.php
index 9f9dd01a93..d3e7698f07 100644
--- a/tests/phpunit/tests/mail.php
+++ b/tests/phpunit/tests/mail.php
@@ -342,4 +342,22 @@ class Tests_Mail extends WP_UnitTestCase {
 			$this->assertEquals( $expected[ $header ], array_pop( $target_headers ) );
 		}
 	}
+
+	/**
+	 * Test that the Sender field in the SMTP envelope is not set by Core.
+	 *
+	 * Correctly setting the Sender requires knowledge that is not available
+	 * to Core. An incorrect value will often lead to messages being rejected
+	 * by the receiving MTA, so it's the admin's responsibility to
+	 * set it correctly.
+	 *
+	 * @ticket 37736
+	 */
+	public function test_wp_mail_sender_not_set() {
+		wp_mail( 'user@example.org', 'Testing the Sender field', 'The Sender field should not have been set.' );
+
+		$mailer = tests_retrieve_phpmailer_instance();
+
+		$this->assertEquals( '', $mailer->Sender );
+	}
 }