The Clarets suffered an early blow when defender Russell Anderson had to be carried off and was replaced by Northern Ireland international Michael Duff.
Chances were always at a premium throughout the entire 90 minutes, but it was midfielder Chris McCann who came as close as anyone as he posed a constant threat to the Fulham defence in and around the penalty area.
It was McCann who drove wide of the target on 13 minutes after receiving a pass from Martin Paterson.
Burnley's best opportunity in the opening half arrived after 20 minutes when Joey Gudjonsson's through ball sent Paterson clear, but he curled his left-foot shot too high from the edge of the box.
At the other end, Fulham's best opportunity arrived midway through the half when Andrew Johnson was sent clear of the Burnley offside trap but shot tamely at keeper Brian Jensen.
McCann saw a shot deflected just wide on the half-hour mark, while prior to half-time Wade Elliott volleyed wide from a Robbie Blake cross.
The second half continued in much the same vain with Burnley enjoying the lion's share of possession and with Fulham posing just the occasional threat.
On one such occasion, Clint Dempsey should have done better than pull his shot wide.
McCann had another great opportunity on 62 minutes but screwed his shot wide after good work by Blake.
And to further illustrate the threat posed by McCann, the Irishman had another opportunity 13 minutes from time but his header lacked power.
With extra time beckoning it was something of a surprise when the deadlock was broken as two of Owen Coyle's substitutes combined with Chris Eagles sending Burnley-born striker Rodriguez clear to round the keeper.