Alfie Haaland, Ian Harte, Danny Granville and Lee Sharpe scored from the spot, while Paulo Sergio and Jorge Soares both missed for Maritimo after the Canadian Alex Bunbury had scored their first.
Graham refused to say whether he would be in charge of Leeds for the second round. When asked, he said: "Who knows. Nothing has changed. But clearly this has to be sorted out."
He had to sort out his team after a disappointing first half, at the very end of which they conceded a goal. That levelled the tie at 1-1 on aggregate.
The first sign of any real danger for Leeds came in the 20th minute when Bunbury was denied by a clearing header from Lucas Radebe. A minute later Haaland was booked, and from the free-kick Soares' header forced Nigel Martyn into a sprawling save to his left.
Leeds eventually managed a long-range pot-shot by Lee Bowyer, but as the pressure mounted towards the interval they cracked, Soares beating Martyn with a fierce near-post header from Adelino Lopes' free-kick.
Nine minutes into the second half Jimmy Floyd Hasselbaink almost took Leeds back in front on aggregate. A well-worked free-kick gave him an opportunity to shoot but his effort flew a yard wide of the left-hand post.
Hasselbaink then had two glorious chances to put Leeds into the next round. A deflected left-foot shot from the edge of the area saw the ball spin narrowly wide of the left-hand post, and within seconds another drive was tipped round the post by Yves van der Straeten.
In the dying moments of the second half, a long throw from the Leeds substitute, Sharpe, bounced its way across the six-yard line before Clyde Wijnhard, another substitute, produced a glancing header which went wide.
In injury-time Wijnhard's cross was bundled goalwards by Hasselbaink with his knee, and Van der Straeten came up with another fine save to send the game into extra time.
In the third minute of the extra period, Maritimo's substitute Tulipa blazed over from 12 yards with only Martyn to beat.
Maritimo: Van der Straeten; Jorge, Soares, J Silva, Jokanovic, Bunbury, Rui Oscar, Gama, Eusebio, Marcio Antonio, Lopes. Substitutes: Tulipa, P Silva, Fernandez, Rodrigues, Vasco, Camara, P Antonio.
Leeds: Martyn; Haaland, Radebe, Hasselbaink, Bowyer, Hopkin, Kewell, Harte, Hiden, Molenaar, Halle. Substitutes: Robinson (gk), Sharpe, Wetherall, Granville, Lilley, Wijnhard, Woodgate.
Referee: F Stuchlik (Austria).Reuse content