Here is the PowerShell code that I came up with. This is intended to work for a site/URL that when you navigate to it, it prompts you to save a document/file.
#Start IE and navigate to your download file/location
$ie = New-Object -Com internetExplorer.Application
$ie.Navigate("http://www.your.website.here.com/awesome_stuff/DOC_1.docx")
#------------------------------
#Wait for Download Dialog box to pop up
Sleep 5
while($ie.Busy){Sleep 1}
#------------------------------
#Hit "S" on the keyboard to hit the "Save" button on the download box
$obj = new-object -com WScript.Shell
$obj.AppActivate('Internet Explorer')
$obj.SendKeys('s')
#Hit "Enter" to save the file
$obj.SendKeys('{Enter}')
#Closes IE Downloads window
$obj.SendKeys('{TAB}')
$obj.SendKeys('{TAB}')
$obj.SendKeys('{TAB}')
$obj.SendKeys('{Enter}')
Please let me know your feedback!!!