Actually FTP seems to be the easiest way to deploy single file. However if you don't want to use it you can use KUDU API. What you need is to wrap it in powershell script:
$username = '$(username)'
$password = '$(password)'
$base64AuthInfo = [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes(("{0}:{1}" -f $username,$password)))
$apiUrl = "https://$(siteName).scm.azurewebsites.net/api/vfs/site/your-file"
$filePath = "$(System.DefaultWorkingDirectory)/your-file"
$headers = New-Object "System.Collections.Generic.Dictionary[[String],[String]]"
$headers.Add("Authorization", ("Basic {0}" -f $base64AuthInfo))
$headers.Add("If-Match", "*")
Invoke-RestMethod -Uri $apiUrl -Headers $headers -Method PUT -InFile $filePath -ContentType "multipart/form-data"
Here you find info how to get credentials.
But if this is one time task you may use drag & drop from KUDU panel as it is mentioned here