Leeds' last victory in the FA Cup came at Crystal Palace in 2003, so long ago that Peter Ridsdale and Terry Venables were at the helm and Harry Kewell scored the winner.
And Leeds, who lacked leading scorer Jermaine Beckford due to a hip injury, were a goal down after only nine minutes.
Former Elland Road striker Leon Constantine got a touch to Ryan Gilligan's pass and veteran striker Scott McGleish, who is on loan from Wycombe, tapped the ball just inside the right-hand post.
Northampton lost injured central defender Chris Doig three minutes later with Alex Dyer his replacement.
Robert Snodgrass, who was making a rare start for Leeds, almost equalised in the 18th minute but his header was cleared off the line by Daniel Jackman.
Leeds' hopes were raised by Coke's sending off for a second yellow card.
He was booked as early as the third minute for fouling Ben Parker and in the 27th minute he brought down Fabian Delph, being watched by new Portsmouth boss Tony Adams.
But Gary McAllister's men were level from the penalty spot ten minutes later.
Jonny Howson was adjudged to have been tripped by Mark Hughes just inside the 18-yard box and Andy Robinson stepped up to fire the spot-kick high into the net.
Leeds had the lion's share of possession after Coke's departure with Robinson smacking a 25-yard shot against the post and Luciano Becchio and Jonathon Douglas also going close.
But Leeds keeper Casper Ankergren was fortunate when McGleish sent a long-range attempt over the bar with the keeper unable to get back in time after leaving his line to narrow the angle for Gilligan.
Leeds hit the woodwork for a second time when the lively Snodgrass saw his shot thud back off the left upright, but they were unable to conjure up a winner.