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 it was Madrid who posed the greater danger early on.
Karim Benzema pulled a shot wide of the far post before his teasing cross was volleyed onto the bar at the back post by Ronaldo.
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.
Seconds later Real were level thanks to a lovely move as Benzema backheeled Luka Modric’s pass into Ronaldo’s path and he poked the ball past Claudio Bravo on the stretch.
The visitors then went on to end the half the stronger as Gareth Bale had a goal ruled out for offside, whilst Bravo tipped a piledriver from Ronaldo over and Bale stabbed inches wide from the resulting corner.
Madrid also had the first opportunity of the second-half as lovely link-up play between Benzema and Ronaldo this time put the Frenchman through on goal, but Bravo did just enough to parry his shot to safety.
And they were again caught by a sucker punch when Alves’s ball over the top was brilliantly controlled and finished by Suarez on the turn.
Ronaldo, who had been booked in the first-half for diving, was then fortunate not to see red when he appeared to kick out after being fouled by Javier Mascherano.
The cautions continued to mount as 11 players were booked in all.
Neymar nearly made amends for his miss in the first period with an incredible run from inside hs own half past numerous Madrid players, but again his finishing was wayward as he blasted over.
Messi had been very quiet given his stunning performances of late, but he was inches away from sealing the points 17 minutes from time as his shot curled just wide.
Neymar then blazed wide again from a Messi pass as space began to open up for Barca.
Those wasted opportunities nearly came back to haunt the Catalans as Bravo had to produce a brilliant save to turn Benzema’s deflected effort behind for a corner.
Casillas kept Madrid in the game at the other end with a pair of fine saves to deny Jordi Alba and Messi, but it was to no avail as Barca racked up their 18th win in 19 games.