diff --git a/backend/swift/swift.go b/backend/swift/swift.go
index b0431aeba..7c1cfbe64 100644
--- a/backend/swift/swift.go
+++ b/backend/swift/swift.go
@@ -25,6 +25,7 @@ import (
 	"github.com/rclone/rclone/fs/operations"
 	"github.com/rclone/rclone/fs/walk"
 	"github.com/rclone/rclone/lib/pacer"
+	"github.com/rclone/rclone/lib/readers"
 )
 
 // Constants
@@ -1224,8 +1225,13 @@ func (o *Object) Update(ctx context.Context, in io.Reader, src fs.ObjectInfo, op
 		}
 		o.headers = nil // wipe old metadata
 	} else {
+		var inCount *readers.CountingReader
 		if size >= 0 {
 			headers["Content-Length"] = strconv.FormatInt(size, 10) // set Content-Length if we know it
+		} else {
+			// otherwise count the size for later
+			inCount = readers.NewCountingReader(in)
+			in = inCount
 		}
 		var rxHeaders swift.Headers
 		err = o.fs.pacer.CallNoRetry(func() (bool, error) {
@@ -1242,6 +1248,10 @@ func (o *Object) Update(ctx context.Context, in io.Reader, src fs.ObjectInfo, op
 		o.md5 = rxHeaders["ETag"]
 		o.contentType = contentType
 		o.headers = headers
+		if inCount != nil {
+			// update the size if streaming from the reader
+			o.size = int64(inCount.BytesRead())
+		}
 	}
 
 	// If file was a dynamic large object then remove old/all segments
diff --git a/backend/swift/swift_test.go b/backend/swift/swift_test.go
index 7b182230a..f7f4e0ecb 100644
--- a/backend/swift/swift_test.go
+++ b/backend/swift/swift_test.go
@@ -2,10 +2,19 @@
 package swift
 
 import (
+	"bytes"
+	"context"
+	"io"
 	"testing"
 
 	"github.com/rclone/rclone/fs"
+	"github.com/rclone/rclone/fs/hash"
+	"github.com/rclone/rclone/fs/object"
+	"github.com/rclone/rclone/fstest"
 	"github.com/rclone/rclone/fstest/fstests"
+	"github.com/rclone/rclone/lib/random"
+	"github.com/stretchr/testify/assert"
+	"github.com/stretchr/testify/require"
 )
 
 // TestIntegration runs integration tests against the remote
@@ -21,3 +30,50 @@ func (f *Fs) SetUploadChunkSize(cs fs.SizeSuffix) (fs.SizeSuffix, error) {
 }
 
 var _ fstests.SetUploadChunkSizer = (*Fs)(nil)
+
+// Check that PutStream works with NoChunk as it is the major code
+// deviation
+func (f *Fs) testNoChunk(t *testing.T) {
+	ctx := context.Background()
+	f.opt.NoChunk = true
+	defer func() {
+		f.opt.NoChunk = false
+	}()
+
+	file := fstest.Item{
+		ModTime: fstest.Time("2001-02-03T04:05:06.499999999Z"),
+		Path:    "piped data no chunk.txt",
+		Size:    -1, // use unknown size during upload
+	}
+
+	const contentSize = 100
+
+	contents := random.String(contentSize)
+	buf := bytes.NewBufferString(contents)
+	uploadHash := hash.NewMultiHasher()
+	in := io.TeeReader(buf, uploadHash)
+
+	file.Size = -1
+	obji := object.NewStaticObjectInfo(file.Path, file.ModTime, file.Size, true, nil, nil)
+	obj, err := f.Features().PutStream(ctx, in, obji)
+	require.NoError(t, err)
+
+	file.Hashes = uploadHash.Sums()
+	file.Size = int64(contentSize) // use correct size when checking
+	file.Check(t, obj, f.Precision())
+
+	// Re-read the object and check again
+	obj, err = f.NewObject(ctx, file.Path)
+	require.NoError(t, err)
+	file.Check(t, obj, f.Precision())
+
+	// Delete the object
+	assert.NoError(t, obj.Remove(ctx))
+}
+
+// Additional tests that aren't in the framework
+func (f *Fs) InternalTest(t *testing.T) {
+	t.Run("NoChunk", f.testNoChunk)
+}
+
+var _ fstests.InternalTester = (*Fs)(nil)