Reading played the last 10 minutes with 10 men after Paul McShane saw red for a dangerous tackle on Kevin McDonald.
After a goalless first half, Reading took the lead when Jordan Obita drilled in off the inside of the post.
Tom Cairney headed Fulham's leveller from close in after Ali Al-Habsi failed to hold on to Scott Malone's drive.
Ryan Fredericks went close to a stoppage-time winner for Fulham but his shot across goal went just wide after a run down the right.
The two sides will meet again in the return leg at the Madejski Stadium on Tuesday.
With the first leg drawn, both sides stretched their impressive unbeaten runs which secured their respective places in the play-offs.
Reading will hope a home record of just two league defeats all season will prove a good omen in Berkshire in the return leg, while Fulham will hope to avenge a 1-0 defeat back in January.
In a tempestuous and rowdy affair in west London, referee Stuart Attwell had a task on his hands to keep a lid on emotions both on and off the pitch.
Reading's goal was perhaps a touch fortunate when an apparent foul by McShane on Stefan Johansen went unpunished and Obita was on hand to finish as the space opened up for the visitors.
As well as McShane's red card, four yellow cards were produced as the sides, who finished sixth and third respectively in the regular season, looked to gain the ascendancy.
Despite the at times overly physical challenges, Attwell showed his composure to ensure McShane's reckless challenge was the only significant punishment for either side.
Fulham manager Slavisa Jokanovic:
"We cannot be completely satisfied with what happens in the game. But I was expecting a tough game and I believed in ourselves to win the game, but the result is what it is.
"Everything is set up for the return on Tuesday and it's open.
"The red card is very clear and I believe it's very clear. It looked a clear free-kick for their goal as there was a touch with the hand too but it's a physical and very complicated game and sometimes referees can make mistakes.
"But we must be ready, we must be better for what's going to come on Tuesday.
"We showed we weren't scared after their goal, they didn't give us a lot of space but we found more in the second half. We had many chances but we weren't clinical enough. We must be strong and believe in ourselves for the return leg."
Reading manager Jaap Stam:
"We can still win a game without Paul McShane, he's been very important to us throughout the season but we have options and we're very confident that we can still do well.
"I'm not going to say who does or doesn't have the advantage for the second leg. We know what we can do and what we can bring as a side and you've seen that today.
"Home advantage doesn't guarantee you a win, you still have to work very hard and give everything for it.
"These games aren't just about ability and quality, it's about your mind and dealing with the stress and playing with your heart. A lot of things are different in these games than they are in the league."
On Paul McShane's red card: "I've seen it back and he hits the player, he's trying to reach the ball and the player has got a touch in there to just nick it in front of him, which is why he's late to the ball.
"We had to deal with it and we knew it was going to be hard, but we defended well and ended the game level.
"Everything is still open and anything can still happen on Tuesday."
Match ends, Fulham 1, Reading 1.
Second Half ends, Fulham 1, Reading 1.
Corner,  Fulham. Conceded by Chris Gunter.
Attempt blocked. Scott Malone (Fulham) left footed shot from the left side of the box is blocked. Assisted by Neeskens Kebano.
Attempt missed. Floyd Ayité (Fulham) left footed shot from the right side of the box is high and wide to the left. Assisted by Tom Cairney.
Corner,  Reading. Conceded by Tim Ream.
Attempt missed. Yann Kermorgant (Reading) right footed shot from the centre of the box misses to the right. Assisted by Tyler Blackett with a cross.
Foul by Scott Malone (Fulham).
Yann Kermorgant (Reading) wins a free kick on the right wing.
Substitution, Fulham. Cyriac replaces Sone Aluko.
Attempt missed. Kevin McDonald (Fulham) right footed shot from outside the box is high and wide to the right. Assisted by Scott Malone.
Substitution, Reading. Tiago Ilori replaces Joseph Mendes.
Paul McShane (Reading) is shown the red card.
Kevin McDonald (Fulham) wins a free kick in the defensive half.
Foul by Paul McShane (Reading).
Sone Aluko (Fulham) wins a free kick in the defensive half.
Foul by Liam Moore (Reading).
Offside, Fulham. Tom Cairney tries a through ball, but Neeskens Kebano is caught offside.
Substitution, Reading. Joey van den Berg replaces John Swift.
Foul by Stefan Johansen (Fulham).
John Swift (Reading) wins a free kick in the defensive half.
Corner,  Fulham. Conceded by Liam Moore.
Foul by Scott Malone (Fulham).
Chris Gunter (Reading) wins a free kick in the defensive half.
Substitution, Reading. Joseph Mendes replaces Lewis Grabban.
Offside, Fulham. Sone Aluko tries a through ball, but Floyd Ayité is caught offside.
Attempt blocked. Sone Aluko (Fulham) right footed shot from the left side of the box is blocked.
Attempt blocked. Stefan Johansen (Fulham) left footed shot from the left side of the box is blocked. Assisted by Kevin McDonald.
Goal!  Fulham 1, Reading 1. Tom Cairney (Fulham) header from very close range to the centre of the goal.
Corner,  Fulham. Conceded by John Swift.
Attempt missed. Yann Kermorgant (Reading) right footed shot from outside the box is high and wide to the left. Assisted by Lewis Grabban with a headed pass.
Substitution, Fulham. Neeskens Kebano replaces Chris Martin.
Tomas Kalas (Fulham) wins a free kick in the defensive half.
Foul by Lewis Grabban (Reading).
Attempt saved. Stefan Johansen (Fulham) right footed shot from outside the box is saved in the top centre of the goal. Assisted by Scott Malone.
Kevin McDonald (Fulham) is shown the yellow card for a bad foul.
Foul by Kevin McDonald (Fulham).
John Swift (Reading) wins a free kick in the defensive half.
Scott Malone (Fulham) is shown the yellow card.
Goal!  Fulham 0, Reading 1. Jordan Obita (Reading) left footed shot from the left side of the box to the centre of the goal. Assisted by Paul McShane.
Fulham and Reading could not be separated as their Championship play-off semi-final first leg finished all square in a fiery encounter.