Jeremy Mathieu headed Barca in front early on from Lionel Messi's free-kick, but Cristiano Ronaldo hit back at his critics in recent weeks with a deserved equaliser before the break.
Suarez was to have the final say, though, as he registered his first goal in El Clasico by controlling Dani Alves's long ball before turning to strike into the far corner.
A third defeat in four games for Madrid will see coach Carlo Ancelotti come under increasing pressure to hold onto his job, but he insisted he had been encouraged by his side's performance for the first hour.
And Ancelotti believes Madrid's title fight is far from over despite dropping eight points in their last four league games.
"Four points is obviously an advantage for Barcelona. We won't give up, particularly because of the way we played in the first-half. We have to keep going to try and play like that for 90 minutes."
However, it was Barca who went in front on 18 minutes when Mathieu headed home Messi's in-swinging free-kick from the left for his first league goal for the club.
Neymar should have doubled the hosts' advantage on the half hour when he fired straight at Iker Casillas with the goal gaping after a mishit Suarez shot rolled across goal.
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