You could try pasting to the same location each time: 
Private Sub CommandButton1_Click()
ActiveSheet.Range("A1:Z10").Copy
Workbooks.Open Filename:="C:\Users\kenneth\Desktop\TEST\OUTPUT.xlsx"
ActiveSheet.Cells(1, 1).Select
Selection.PasteSpecial Paste:=xlPasteAll, Operation:=xlNone, Skipblanks:=False, Transpose:=True
ActiveWorkbook.Save
ActiveWorkbook.Close
Application.CutCopyMode = False
End Sub
but be aware that it copies the whole range and overwrites the previous version. Would that do?