diff --git a/index.js b/index.js
index 5a7dfd2..45165ef 100644
--- a/index.js
+++ b/index.js
@@ -29,7 +29,11 @@ function doRead(handle, length) {
 	}).then((result) => {
 		if (result.bytesRead === 0) {
 			// Not documented; https://github.com/nodejs/node/issues/35363
-			throw new EndOfStream();
+			return Promise.try(() => {
+				return handle.close();
+			}).then(() => {
+				throw new EndOfStream();
+			});
 		} else if (result.bytesRead === length) {
 			return buffer;
 		} else if (result.bytesRead < length) {