American defender Bocanegra thumped home a superb goal from a Franck Queudrue free-kick just two minutes after John Carew had given Villa the lead.
But Chris Coleman's men couldn't finish the job despite having the better of the chances for the rest of the match.
Michael Brown twice hit the woodwork and Brian McBride went close on three further occasions, but Villa - with Gareth Barry outstanding - almost nicked it in injury time.
Carew, who could have joined Fulham earlier in his career before failing a medical, turned in the area - only for Niemi to produce a superb left-handed save.
In a frustrating match, Fulham just about had the better of the first half, which at best was bitty and at worst scrappy.
The home side went closest to opening the scoring early on when livewire Brown, who never stopped running all match, hooked a left-foot volley from outside the box towards goal.
His effort deflected off Barry and cannoned against the crossbar with Villa keeper Thomas Sorensen completely wrong-footed.
Queudrue also had a long-range effort well saved as Fulham, still without suspended striker Vincenzo Montella, surged forward.
But they were undone after 20 minutes when referee Mason unwittingly played a part in Villa's opener.
Firstly, he refused to give a foul when Alexei Smertin was left on the floor in midfield and then, when Fulham cleared the danger, he got in the way of the Russian midfielder as he tried to receive a pass.
Barry took full advantage, running into the area and setting up Carew with a slide-rule pass. The big striker turned and fired a shot which trickled past keeper Niemi and into the net.
If Fulham were furious they made that emotion count because right from the kick-off they fashioned an equaliser.
Smertin won a free-kick on the right and when Queudrue curled the ball over, Bocanegra arrived unmarked to thump home a spectacular diving header.
Fulham could have taken the lead when Martin Laursen's terrible pass was snapped up by Tomasz Radzinski after 35 minutes and he sent Brown away. But Sorensen bravely saved at the midfielder's feet and McBride's follow-up effort was fluffed.
In the second half Fulham continued to edge the play, without finding a killer instinct.
McBride drove in three shots in the space of ten minutes, two of them being saved. And Brown came mighty close to making it 2-1 when he latched on to one rebound but shot against the post.
Simon Davies also had a volley saved for the hosts while Villa almost snatched it at the end. But a draw was a fair result.