Took your code as a basis (THANKS!!!) but had to modify it to make it work.
It didn't handle multiple rows, all cells were put after each other.
2 loops: one to go through the rows and one to go through the cells of each row.
Each time the row loop starts the temporary string is emptied. Before starting a new row, the temp string is added to the Outfile.
Sub ToCsv()
Dim rng As Range
Dim row As Range
Dim cell As Range
Dim ItemNew As String
Set rng = Range("A1:E2") 'Adjust the range accordingly
OutFile = FreeFile
Open "Outfile.csv" For Output As #OutFile
'Print #OutFile, "Header"
For Each row In rng.Rows
ItemNew = ""
For Each Item In row.Cells
If InStr(1, Item, Chr(34)) > 0 Then Item = Chr(34) & Replace(Item, Chr(34), Chr(34) & Chr(34)) & Chr(34)
If InStr(1, Item, ",") > 0 And Left(Item, 1) <> Chr(34) Then Item = Chr(34) & Item & Chr(34)
If InStr(1, Item, vbLf) > 0 And Left(Item, 1) <> Chr(34) Then Item = Chr(34) & Item & Chr(34)
If ItemNew = "" Then
ItemNew = Item
Else
ItemNew = ItemNew & "," & Item
End If
Next
Print #OutFile, ItemNew
Next
Close OutFile
End Sub