Paul Lambert's side had been unlucky not to have led at the break but Charles N'Zogbia finally made the breakthrough before the hour mark at Villa Park.
However, Delph's own goal, combined with a glaring miss in the 73rd minute, allowed Fulham to go home with a point and denied Villa the chance to go five points clear of the bottom three.
Villa headed into the match seemingly timing their pull from relegation perfectly, winning three of their past four to leave them two points clear of the drop zone heading into the game, with rivals Sunderland and Wigan not in action.
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.
In the 31st minute Benteke and N'Zogbia played a series of one-two passes with each other before the former found his colleague with a delicate cross on the left, which N'Zogbia tried to clip home only for Mark Schwarzer to save impressively.
Villa kept up the pressure from the restart and it 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. Bryan 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.
Villa had a sublime chance in the 73rd minute 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.