Federici went up for a corner which was cleared off the line by Cardiff defender Gabor Gyepes, only for the keeper to thunder the ball into the net.
"He fancies himself as an outfield player," said Reading manager Steve Coppell. "We played him out on the field in a pre-season friendly against Bromley and he scored.
"It was just instinct that got me up to poke it in," said Federici. "I've worked out a special celebration for when I score but I never got the chance to do it because everyone jumped on me. I will have to save it for the next one I get."
His equaliser sparked angry scenes at the Cardiff end of the ground and after the game mounted police were called in to maintain order. Welsh supporters had been in full song after believing Cardiff had grabbed a last-gasp winner from Chopra.
There was only a minute of normal time left when Chopra, who looked to be in an offside position, shot home from a tight angle to celebrate Dave Jones' 600th game in charge as a league manager.
There was controversy about that goal and referee Andy Hall was the unwanted centre of attraction throughout, booking four players in a feisty first half, but angering both sets of supporters with many of the decisions he did, or sometimes didn't, give.
"Four minutes of extra time were signalled, but then he added on another minute-and-a-half. I don't know where he got that from," said angry Cardiff boss Jones. "His changing room was next to our dressing room and after the game he must have found out what I thought of him through the wall."
An early booking for Reading's Noel Hunt set the pattern and next to see a yellow card was Cardiff's Chopra.
Reading's Aussie keeper Federici bravely dived at Chopra's feet as they went for a Wayne Routledge throughball. Both players were injured, Federici with a damaged shoulder and Chopra with a leg injury.
They were able to continue after treatment, although Chopra was booked for his challenge.
Federici who is in the side while Marcus Hahnemann recovers from a calf injury, was being watched by his father and uncle, who had flown in from Australia to watch him play.