The Merseysiders protested about some of the decisions that did not go their way, and arguably had a case with Jon Flanagan's second-half sending off, but there seemed little doubt about the award of two penalties, that were both converted by Cole.
With United continuing to create the better chances, Cole smashed in his third to set the seal on a convincing victory for the home side to the delight of almost 3,000 fans.
In front of the watching Sir Alex Ferguson, the Reds made an instant impression but the opening quarter of an hour became a rather cagey affair. Captain Tom Thorpe made a couple of important early blocks and formed a formidable barrier at the heart of the defence with Michael Keane.
A glorious pass by Adnan Januzaj, bent with his left foot into the path of Tom Lawrence, created the opener with Stephen Sama bringing down the Welshman. Sama escaped with a yellow card, having been sent off in both of the league fixtures against United this term, and Cole calmly ignored some blatant gamesmanship from Adam Morgan to send Danny Ward the wrong way from the spot.
Seven minutes later, advantage was played when Conor Coady fouled Charni Ekangamene but Jon Flanagan then shoved Lawrence inside the box. Cole took on the responsibility again from 12 yards, despite Jesse Lingard offering to take the penalty, and made the right call as he nervelessly found the opposite corner of the net with supreme coolness.
There were few clear-cut chances created in open play in the first half and even Morgan's wild