Luis Enrique's side started brightest with two good chances for Lionel Messi and Neymar, before Fernando Torres gave the visitors a shock lead before being dismissed in a dramatic opening 45 minutes.
It was an explosive start to the second period with multiple chances from Messi and Neymar in the opening ten minutes of the half, before Suarez clinched the equaliser just after the hour mark. He then bagged his second with 17 minutes to play, which proved to be the crucial difference.
The rightfully named battle of the unstoppable force in Barcelona against the immovable object in Atleti pre-match proved to be the case in the opening proceedings.
Messi got his eye on goal in the 13th minute when Jordi Alba did well to get a floated cross into the Argentine at the far post but, despite it been on his favoured left foot, Messi volleyed well wide of the mark.
Neymar should have given the home side the lead six minutes later when Messi picked out Dani Alves who teed it up to his compatriot with the header, but he managed to head over from seven yards.
Hero to Zero
Torres has looked the shadow of the player that he was in his early Liverpool days but managed to pull out a moment to open the scoring after a quarter of an hour.
It was poor defending from Barcelona's point of view as Koke was allowed to pick out of the Spaniard, and he made no mistake as he coolly stuck it between the legs of Marc-Andre ter Stegen to send the Nou Camp into a stunned silence.
It should have been two just after the half-hour mark after a counter-attack, with Torres playing it through to Antoine Griezmann whose stretched effort looked to be creeping in, but ter Stegen managed to keep it out with a smart save.
Diego Simeone and his side were left in shock nine minutes from the break though as goalscorer Torres was dismissed. It won't come as a surprise to many though as two crude challenges on Neymar and Sergio Busquets respectively saw him walk and left his side with a mountain to climb.
Barging down the door
Enrique's side looked a whole different animal as the sides entered the pitch for the second period as the combination of Neymar and Messi had five great chances to grab an equaliser.
The first came from Messi as his attempted overhead kick went just wide, but Neymar came much closer seconds later as he looked to have beaten Jan Oblak only to clatter the crossbar with his curling effort.
Messi looked to have slithered through the Atleti defence in the 51st minute but Filipe Luis did excellently to deny him a equaliser as a defensive block took it off the toe of the Argentine.
Neymar followed that with another headed effort as Ivan Rakitic put a great ball into the area, but he faltered once again as it was straight into the arms of Olbak.
The bombardment seemed never ending as two more chances followed stemming from poor defensive errors from Simeone's side. Both Messi and Neymar were allowed to cut inside, but luckily for them both were short of the target.
After such a persistence, Atleti's defence was eventually breached just after the hour mark with Alves' ball into the area just missing Neymar, but falling perfectly for Alba.
The Spaniard looked to have dragged his shot wide but for Suarez, who had been quiet up to that point, to turn it home from close-range.
Atleti had been on the ropes for sometime but as the game entered the final 20 minutes they looked turned over as Enrique's side took the lead for the first time.
It was a great piece of play from a section of South American contingent as Messi played it off to Suarez who quickly played it out wide to Alves.
The Brazilian had proven a menace all evening and he was at it again as his cross was just too much for Diego Godin, but it was perfect for his compatriot Suarez who powerfully placed his header beyond Olbak to give Barca the advantage heading to the Vicente Calderon next Wednesday.