The Clarets were punished when the Colombian star headed home deep into injury time.
Brian Laws' men, who had never won at Wigan, piled on the pressure in the early stages with player-coach Graham Alexander having a shot blocked and Steven Fletcher forcing Chris Kirkland into action.
In the fourth minute David Nugent flashed a shot over the bar from the edge of the box and soon after Gary Caldwell brilliantly blocked Wade Elliott as he was about to pull the trigger.
But Wigan were denied the lead in the 18th minute by the frame of the goal when Caldwell rose highest to meet James McCarthy's corner, but his bullet header crashed against the post.
Burnley were again the better side after the break with Nugent fluffing another good chance wide.
On the hour Alexander curled a free-kick inches wide after Fletcher had been fouled, with Marcelo Moreno coming closest for the below-par home side.
In the 73rd minute the woodwork came to Wigan's rescue and denied Martin Paterson a stunning goal.
Nugent's ball over the top found the Northern Ireland international and his first-time volley cannoned back off the bar with Kirkland beaten.
Wigan finally came to life in the closing stages with Brian Jensen forced into action four minutes from time as he tipped over Jason Scotland's shot.
And the giant Dane was at it again a minute later when he somehow kept out Mohamed Diame's fierce volley as Burnley began to rock.
Wigan's late pressure finally told at the death though when Victor Moses' brilliant cross found Rodallega to launch himself into a diving header that flew past Jensen and broke the visitors' hearts.
