The Spiders' David Galt headed against the outside of the post before Shire took the lead.
Thomas Orr took advantage of Bryan Wharton's poor header to burst through on goal and finish past William Muir.
David McKenna doubled the advantage from close range and Scott Ferries' effort completed the win.
East Stirlingshire moved off the bottom of Scottish League Two with a surprise victory over promotion-chasing Queen's Park.