The other answer does get you the response, but you need an additional step to get the actual body of the response, not just the headers. Here is a snippet:
try {
$result = Invoke-WebRequest ...
}
catch {
$result = $_.Exception.Response.GetResponseStream()
$reader = New-Object System.IO.StreamReader($result)
$reader.BaseStream.Position = 0
$reader.DiscardBufferedData()
$responseBody = $reader.ReadToEnd();
}