Away supporters called for manager Felix Magath to be sacked as the Cottagers slumped to a fifth defeat in six Sky Bet Championship games.
The visitors finished with 10 men, adding to the pressure on manager Magath who only joined in February.
Reading won at a canter, though Fulham’s cause was rendered almost lost as early as the 18th minute, when former Leeds striker Matt Smith, also on debut, was sent off for a lunge on Hope Akpan.
The west London club endured a miserable end to last season, winning only three matches out of 12 under Magath as they were relegated from the Premier League.
Fulham have started the Championship similarly poorly, losing their opening four matches before earning their first point in a 1-1 draw with Cardiff.
It hardly got any better for them at the Madejski Stadium, where they were a goal down and a man down before having the chance to settle into the game.
Fulham had begun brightly, with Ross McCormack volleying over from close range after a Smith header.
But they went behind in the 15th minute from Reading’s first meaningful attack, led by Jordan Obita along the left flank.
Obita crossed early and Murray, on loan from Crystal Palace until January, dived forward to head in from close range for his first goal for the club.
Three minutes later, it got worse for Fulham. Smith went in late and high on Akpan and referee Mike Jones issued the red card immediately.
Fulham responded well, with Tim Hoogland denied only by a last-ditch tackle from Chris Gunter and Chris David forcing a fine save from home goalkeeper Adam Federici.
But Reading swiftly assumed control, with Murray and Gunter going close from well-struck efforts.
The Reading pressure continued after the interval, with Simon Cox volleying straight at Fulham goalkeeper Gabor Kiraly.
Kiraly, on his debut, was in action soon after, making a stunning save to keep out a powerful strike from Jake Taylor.
But the pressure inevitably told, with a second goal arriving in the 54th minute when Murray nodded in a precise corner from Obita from close range.
It was too much for the 2,745 Fulham fans, who voiced their displeasure at manager Magath.
“You’re getting sacked in the morning” was followed by further choruses of “Felix out”.
The visitors could have been swamped. Kiraly tipped a header from home substitute Pavel Pogrebnyak – a former Fulham player – against the crossbar and Taylor drove narrowly wide from distance.
But they could not hold out at 2-0 until the end. In the 85th minute, substitute Blackman ran through, went by Kiraly and slotted the ball home. Fulham’s misery was complete.