Fulham started brightly, with Cauley Woodrow combining well with Pajtim Kasami before the latter's header was sent over the crossbar.
Villa had an opportunity to take the lead thanks to Leandro Bacuna, but his chipped cross was headed wide by Andreas Weimann, who had found space at the back post.
Bacuna continued to look dangerous with the ball at his feet, and he was close to finding a goal of his own when his low effort was pushed away by David Stockdale.
Woodrow was given one more chance to open his account before the break after intercepting a Villa pass, but the young forward's strike from a tight angle was unable to beat Brad Guzan.
The opening stages of the second half was played in open fashion, but a lack of chances frustrated both sets of supporters.
Fulham took the lead in the 61st minute through Kieran Richardson, who took advantage of some space from outside the box to fire a fine strike past the helpless Guzan.
However, the visitors were unable to hold on for too long as Villa hit back just nine minutes later when Grant Holt arrived at the back post to head in a delivery from a corner.
The two goals set up a dramatic finish to the match, with Fulham coming agonisingly close to regaining the lead when Matthew Lowton cleared Lewis Holtby's effort off the line.
For the second time in the afternoon, Weimann missed a good chance to beat Stockdale as he headed Marc Albrighton's delivery over the bar.
There proved to be one final twist to the game in the 86th minute when substitute Rodallega found space to head Holtby's inch-perfect cross past Guzan and secure all three points.
Source: www.sportsmole.co.uk
