Without club captain Son Heung-min, who has joined up with South Korea before the Asian Cup, the hosts failed to convert a number of chances before the breakthrough in the 78th minute.
Burnley goalkeeper Arijanet Muric threw the ball out, but Porro won a challenge and then arrowed a superb shot into the top corner from 25 yards out for what proved to be a wonderful first goal of the season since his permanent arrival from Sporting Lisbon.
The Clarets' best chance fell to Zeki Amdouni in the first half when it was goalless, but he could only shoot over from a central position inside the penalty area.
In a thrilling finish, Muric twice went up for Burnley corners and from the second one, in the seventh minute of added time, won a near-post flick-on and his header fell to an unmarked Amdouni, who could not hit the target from six yards with a volley on the turn.
The defeat, the Clarets' fifth in seven matches in all competitions, leaves Vincent Kompany's side - 19th in the Premier League - to focus on trying to avoid relegation.
The draw for the fourth round of the FA Cup will take place at 19:50 GMT on Monday, 8 January, before the Wigan Athletic v Manchester United match.
-- Courtesy of BBC Sport