The Uruguayan couldn't inspire his side to victory as the hosts came from behind to win the first Clasico of the season
The Uruguay star enjoyed a dream start to life in a Barca shirt as he laid on Neymar's fourth-minute opener, but Madrid looked to hit back immediately as Karim Benzema missed two good chances soon after.
Lionel Messi's customary composure in front of goal then deserted him before his great rival Cristiano Ronaldo levelled from the penalty spot before half-time.
That was the first league goal conceded by Barcelona this season but Pepe headed Carlo Ancelotti's team ahead and there was a party atmosphere at the Santiago Bernabeu when Benzema converted a brilliant 61st-minute counter-attack.
Real close to within a point of their historic foes at La Liga's summit, while Sevilla can go level with Barca if they beat Villarreal on Sunday.
The pre-match announcement of Suarez's inclusion from the start - one of five Barcelona changes - set pulses racing and the Uruguayan did not disappoint in making an instant impact.
His floated pass to the left flank picked out Neymar, who cut inside past Dani Carvajal and Pepe to find the bottom right corner.
Javier Mascherano and Gerard Pique were caught napping as Real sought an immediate response - Benzema's resulting effort touched wide by Claudio Bravo before Marcelo misplaced a 10th-minute cross behind the France striker with the goal gaping.
There was further frustration for Benzema when a header from Ronaldo's left-wing cross thudded against the crossbar and he lashed the rebound over.
Messi, who picked up a yellow card for an early foul on Toni Kroos, passed up a glorious opportunity to equal Telmo Zarra's all-time Liga scoring record when he inexplicably touched Suarez's teasing centre wide from close range in the 23rd minute.
Carvajal - brought back into Real's backline alongside the returning Sergio Ramos - then produced a vital challenge to deny Neymar and Real were level 10 minutes from half-time.
A prone Pique blocked Marcelo's low cross with his arm, leaving the referee with little option and Ronaldo sent Bravo the wrong way to maintain his record of netting in every league game this term.
Barca almost suffered further damage as Clasico debutant James Rodriguez stole in to head wide from another Marcelo delivery.
Pique turned a Ronaldo effort behind in the 50th minute but returned to his poor form of the opening period, weakly allowing Pepe far too much room to bury a header from the resulting corner.
Jeremy Mathieu - a surprise selection at left-back ahead of Jordi Alba - forced a sprawling Iker Casillas to touch a long range effort behind as Barca looked to redress the balance but Benzema had the goal his endeavours deserved after the hour.
Real streamed forward from a Barcelona corner, with Isco stealing between an unusually ponderous Andres Iniesta and Mathieu for Ronaldo and Rodriguez to work possession toward Benzema, who this time made no mistake.
Chasing the game, Barcelona found themselves at the mercy of Real's relentless brilliance on the break. They were fortunate to suffer no further damage to the scoreline and only severely dented pride at their hands of their bitterest rivals.