The Catalan club also benefited from an own goal by Madrid defender Raphael Varane.
Another setback at home on Saturday will practically end Madrid's title hopes in the league, as it trails Barcelona by nine points.
Former Real Madrid and Spain coach Vicente del Bosque says he believes in the capacity of Santiago Solari to manage Real Madrid, adding that that "he has been doing well since the beginning". They didn't take their chances in the first half but we took ours later. La Liga's scheduling giving Real Madrid a day less to prepare for both Copa semifinal legs has also been an issue for many at the Bernabeu, with Solari calling the calendar "curious" last weekend.
Their surge is timely given a second leg against Lyon in the Champions League is just around the corner, with the tie goalless and delicately poised heading into the return at the Camp Nou. It would leave the Champions League as the only competition in which it can still succeed in its first season without Cristiano Ronaldo in almost a decade.
"He is a very good player, he is not Cristiano or Messi, he is not regular, he always plays well and there is more pressure than at Tottenham". The hosts created most of the scoring chances, including a couple of clear ones for young Brazilian forward Vinicius Junior.
Yet Barca went ahead against the run of play in the 50th minute as Ousmane Dembele led a counter-attack and Suarez slotted his pass into the net.
Although Reguilon caught the eye and Vinicius occasionally dazzled, the biggest difference-maker from an attacking point of view was Ousmane Dembele, who broke free of the Madrid defence to deliver two ideal assists and sentence the tie for good.
It's not too often we see a Barca player compliment a rival who plays for Madrid, but Alba clearly recognises talent when he sees it.
Barca have now scored at least three times in victories on four of their last five visits to the Bernabeu across the Copa and La Liga, but Solari denied that this result would have a big impact on his team's future games, starting with Saturday's Liga Clasico back at the Bernabeu.
Gareth Bale is far from the darling of the Bernabeu these days, and started on the bench in the Copa Del Rey on Wednesday.