Charles N'Zogbia broke the deadlock for Paul Lambert's side ten minutes into the second half with a wonderful curler.
But the hosts' defensive frailties saw their run without a clean sheet extended to 21 outings when the unfortunate Fabian Delph headed into his own net from a corner.
Fulham suffered a blow in the 10th minute when Damien Duff was forced off with a leg injury and replaced by Stanislav Manolev in an otherwise quiet opening period.
Ashley Westwood's 24th-minute corner caused Fulham a moment of anxiety before Christian Benteke side-footed over from towards just outside the six-yard box when he should have added to his overall tally of 19 goals from Andreas Weimann's ball in.
N'Zogbia saw a left-footed shot straight at Mark Schwarzer parried over and Andreas Weimann sent a half-volley wide shortly before half-time as Fulham reached the break grateful to be on an even keel.
Villa kept up the pressure from the restart but they were nearly punished on the counter in the 50th minute when Bryan Ruiz's superb pass found a seemingly offside Dimitar Berbatov, only for the Bulgarian to bend his low shot onto the outside of the left-hand post.
Villa's pressure finally told in the 55th minute. Matthew Lowton found Weimann in the box and though Brede Hangeland blocked the Austrian's attempt, N'Zogbia had time to shift the ball onto his left foot and bend it into the left-hand corner.
The equaliser came in the 66th minute. Ruiz's corner from the right should have been dealt with comfortably at the near post but Villa midfielder Delph contrived the flick the ball backwards with his head, leaving Brad Guzan rooted and Westwood unable to scramble it away before it crossed the line.
The hosts had a good chance late in the game when Delph wasted the opportunity to atone for the own goal, blasting a loose ball at the back post high and wide with his left foot from Weimann's teasing cross.
The point did help Villa move three points clear of the drop zone with five games remaining while mid-table Fulham reached the 40-point mark.