The tricky midfielder popped up in the first of three added minutes to cap a slick passing move with a neat finish, bringing a silver lining to a largely gloomy game at Turf Moor.
With the championship already secured, Ole Gunnar Solskjaer and Warren Joyce used the second of two dead-rubber league games to again blood a string of youngsters from Paul McGuinness' under-18s. Cole joined Michael Keane, Jessie Lingard and Etzaz Hussain in starting for a youthful second string.
United's promising start to the game belied the scrappy affair which followed. Cameron Stewart, moonlighting as a lone striker once again, seized on a defensive mistake and strode through on goal, but his early shot was turned away by Diego Penny. The Peruvian goalkeeper would not face another shot on target before half-time.
Neither Penny nor United counterpart Ron-Robert Zieler were employed during a first 45 minutes of misplaced passes as both sides struggled to find a telling final ball.
Burnely's cohesion increased towards the end of the first period, and Oliver Norwood was slightly fortunate not to concede a penalty for clipping Wes Fletcher's heels. From the rebound, Ray Harvey forced Zieler into his first save of the half.
The first chance of the second period fell to Stewart again as he raced onto Lingard's pass and attempted to round Penny, only for the giant goalkeeper to thrust out a hand and avert the danger.