That's what WinSCP does by default. Make sure you have not turned off the "Preserve timestamp" option:
https://winscp.net/eng/docs/ui_transfer_custom
And of course, the server must support preserving the timestamp.
If you need further help, please post a session log file.

When Copy a file to remote, the file appears with the current date.
It is possible to keep the file original date?