Fulham took the lead from a Sean Davis goal and hung on desperately despite determined pressure from the Blues until substitute Jeffers popped up in the 90th minute to grab the first goal of his second spell at Everton and set up a replay.
Duncan Ferguson headed goalwards from a Thomas Gravesen corner, Edwin Van der Sar parried, but it fell at the feet of Jeffers who pushed the ball home from six yards.
That Everton were behind was against the run of play. They started brightly and dominated throughout the game.
Their first effort came with just a minute on the clock when Kevin Kilbane released Tomasz Radzinski on the edge of the box, but the little Canadian fired over the bar.
The visitors responded in the seventh minute when Luis Boa Morte picked up on a loose Ferguson ball. The Portuguese forward raced to the edge of the box, but his shot did not match his turn of pace and it was easily dealt with by Nigel Martyn.
But it was Everton who were on top and created chance after chance but they just couldn't find the back of the net.
Ferguson came closest when Kilbane crossed from the right and the big Scot rose high in the air to power an effort at goal, but Moritz Volz was on hand to clear the ball off the line.
Recalled Alex Nyarko was next up with a 25-yard screamer. A deflected Alan Stubbs free-kick fell to him just outside the Fulham penalty box. He controlled, turned and launched the ball, but it went sailing over the bar.
And in the last five minutes of the first half the home side had five more shots all wide of the mark, but after the restart Fulham showed their hosts how it should be done.
Four minutes in, it was the man once targeted by David Moyes who grabbed the lead.
But for a failed medical Sean Davis could well have been playing in a blue shirt but instead he was on hand to fire Fulham into the lead.
He started the move with a 20-yard shot which was blocked on the line and eventually cleared by Tony Hibbert, but only as far as Davis running into the six-yard box to fire home.
After that the Cottagers defended desperately and hung on until the 90th minute, repelling wave after wave of Blues attacks until Jeffers grabbed the equaliser.