Thanks to alex and Kyle's hints I looked up to these 2 posts which solved my problem.
nb -> no cookie handling and Fiddlr body copy of the http POST request the 2 required steps for me
here it is my solution:
Sub eds()
Dim strCookie As String, strResponse As String, _
strUrl As String
Dim xobj As Object
Dim WinHttpReq As Object
Set xobj = New WinHttp.WinHttpRequest
UN = "2myusername"
PW = "mypass"
strUrl = "http://my.website.com/"
xobj.Open "POST", strUrl, False
xobj.SetRequestHeader "User-Agent", "Mozilla/5.0 (Windows NT 6.1; WOW64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/33.0.1750.154 Safari/537.36"
xobj.SetRequestHeader "Content-Type", "application/x-www-form-urlencoded"
xobj.Send "username=" & UN & "&password=" & PW & "&login=login"
strResponse = xobj.ResponseText
strUrl = http://my.website.com/date=09-10-1945%&search 'the url pointing to the CSV file
xobj.Open "GET", strUrl, False
xobj.SetRequestHeader "Connection", "keep-alive"
xobj.SetRequestHeader "User-Agent", "Mozilla/5.0 (Windows NT 6.1; WOW64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/33.0.1750.154 Safari/537.36"
xobj.Send
strCookie = xobj.GetResponseHeader("Set-Cookie")
strResponse = xobj.ResponseBody
If xobj.Status = 200 Then
Set oStream = CreateObject("ADODB.Stream")
oStream.Open
oStream.Type = 1
oStream.Write xobj.ResponseBody
oStream.SaveToFile "C:\Users\me\Desktop\ciao\file.csv", 1
oStream.Close
End If
End Sub