Barca even had the luxury of leaving Lionel Messi on the bench, but the Argentine did make his return from a two-month injury layoff half an hour from time and played a part in Suarez's second and Barca's fourth goal yesterday.
Suarez got Barca off to the perfect start after just 10 minutes and Neymar deservedly doubled the visitors' advantage just before half-time.
At that point Neymar and Suarez had scored every one of Barca's last 19 La Liga goals, but Andres Iniesta ended that run in style when he blasted a third into the top corner.
"I didn't see it going like that," said Barca boss Luis Enrique.
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"We were at a superior level and generated superiority in many areas of the pitch. We were solid at the back too and it was more the merit of Barca than the fault of Madrid."
Despite just two defeats in 16 games in charge, Madrid boss Rafael Benitez's job is now in doubt as the home fans turned on club president Florentino Perez towards the end of the game.
"What worries me is the team recovering mentally before the next game."
The build-up to the match had been dominated by security fears after last week's series of deadly attacks in Paris which killed 130 people.
A minute's silence was held to the tune of the French national anthem La Marseillaise and a large French flag was unfurled by the Madrid fans before kick-off in honour of the victims before kick-off.
However, the sombre atmosphere quickly turned hostile as the home fans saw their side completely outplayed by Barca.
Karim Benzema started for Madrid for the first time since he was charged with complicity to blackmail and conspiring to commit a criminal act in his native France in a case involving a sex tape featuring French international teammate Mathieu Valbuena.
And after six weeks out, Benzema's rustiness showed as he failed to make a clean contact when presented with a great chance from Jeremy Mathieu's slip.
It could even have been three before the break as Marcelo cleared Suarez's shot off the line and Raphael Varane blocked Ivan Rakitic's follow-up effort.
The hosts started the second period slightly better as James Rodriguez forced Claudio Bravo into a brilliant save low to his right.
But any hope of a comeback was emphatically ended by a brilliant strike by Iniesta as he latched onto Neymar's backheel eight minutes after the break.
To add to Madrid's woes, Enrique then introduced Messi.
Cristiano Ronaldo was largely anonymous, but he had the chance to briefly restore some pride for Real only to see Bravo produce another fine save.
Messi got in on the act when his pass was flicked-on by Neymar and Suarez took his time before chipping over a helpless Navas.
And Madrid's misery was complete when Isco was shown a straight red card for lashing out at Neymar five minutes from time.