Playing for the first time in a competitive game since being handed a fourth month ban for biting Giorgio Chiellini at the World Cup, Suarez's Barça career had got off to a flying start as he set up Neymar to slide home the opening goal after just three minutes.
However, Real responded like European champions as Cristiano Ronaldo levelled from the penalty spot before second-half goals from Pepe and Karim Benzema handed them a deserved ninth consecutive victory in all competitions.
Barça boss Luis Enrique had sprung a few surprises with his starting line-up as not only did Suarez start, but Xavi and Jeremy Mathieu were also preferred to Ivan Rakitic and Jordi Alba.
However, the Uruguayan proved his worth within three minutes as his cross-field pass was taken down by Neymar who skipped inside Dani Carvajal and Pepe before sliding the ball into the far corner for his 11th goal in as many games this season.
The goal sparked Madrid into life as Benzema failed to find the target from a narrow angle as he was pressured by Claudio Bravo before the Frenchman crashed a header off the bar from Ronaldo's cross.
Both Messi and Neymar were booked for late challenges on Toni Kroos and Carvajal as the hosts upped the pressure in search of an equaliser.
However, they were thankful to captain Iker Casillas midway through the half as he spread himself to stop Lionel Messi becoming the joint top goalscorer in La Liga history from a tempting cross from Suarez.
Just when Barça had appeared to have taken the sting out of Madrid's pressing, though, they gifted the hosts a route back into the game as Marcelo's low cross was blatantly handled inside the area by Gerard Pique.
Ronaldo stepped up and confidently buried the resulting spot-kick to become the first man to score against Barcelona in nine league games this season.
The home side could even have gone in at half-time in front as James Rodriguez's header drifted just wide from another Marcelo cross. It took Carlo Ancelotti's men just five minutes to go in front after the break as Toni Kroos's out-swinging corner was powered past Bravo by Pepe.
Barça came close to opening up the Madrid defence a couple of times as Suarez slipped after turning Sergio Ramos in the box before Ramos made a last-ditch tackle to prevent Messi turning home Neymar's cut-back.
Casillas then produced another stunning save to prevent Jeremy Mathieu's rising drive finding the top corner.
However, in their haste for an equalizer, Barça were caught on the counter-attack from their own corner as Isco, Ronaldo and Rodriguez combined to play in Benzema to fire home his seventh El Clasico goal.
Suarez's debut was then brought to a premature end as he made way for Pedro Rodriguez and matters got even worse for the visitors when Andres Iniesta hobbled off to be replaced by Sergi Roberto.
And Madrid comfortably saw the game out in the final 20 minutes to inflict Barça's first league defeat of the season.