Luis Enrique's side were still smarting from a 2-1 Clasico defeat to Real Madrid on Saturday, and Camp Nou was stunned into silence once more when Fernando Torres struck the opening goal of the game - and his 11th in 15 games against Barca - with Atletico's first shot on target after 25 minutes.
Torres made preserving his side's advantage far more difficult when he picked up a needless second booking just 10 minutes after scoring, and Barca began to turn up the pressure, with Lionel Messi close to converting a fine overhead kick and Neymar rattling Jan Oblak's crossbar.
Atleti's admirable resistance was eventually broken when Suarez steered home Jordi Alba's effort on 63 minutes, and the Uruguay striker finished off a fine team move 16 minutes from time to give Luis Enrique's side the lead with his seventh goal in five home Champions League games this season.
Diego Simeone's team have now lost seven games in a row against the European champions but will still consider themselves to be in with a chance of reaching the last four when the two reconvene at the Vicente Calderon next Wednesday.
Barca created the best early chance when Neymar headed over the crossbar from six yards out after rising highest to meet Dani Alves' cross from the right.
Antoine Griezmann, deployed in a more central role, dragged a shot just wide of the near post, but Atletico were much more clinical moments later.
Koke prodded the ball behind Gerard Pique - who had stepped out to close the midfielder down - into the path of Torres, who struck a low shot home through the legs of Marc-Andre ter Stegen.
Griezmann was denied a second for Atleti as Ter Stegen got fingertips to his bouncing low effort from the edge of the area, but an increasingly frustrated Camp Nou was given a lift when Torres, who had earlier hacked down Neymar, was given a second yellow card for bundling over Sergio Busquets from behind.
Barca looked to up the tempo early in the second half. Messi chested down Neymar's cross before steering an overhead kick just wide of the far post, before the Brazilian curled an effort onto the crossbar from just inside the area.
Messi, subdued in the first half, looked far brighter against Atleti's 10 men, and he forced Oblak into a good low save to his left shortly after Filipe Luis did brilliantly to hook the ball away from the Argentinian's feet from two yards out.
Neymar bent another shot a whisker wide after cutting in from the left, but just as Barca were beginning to look frustrated once more, they found a fortunate equaliser. Alba miscued a volley from Alves' cross straight into the path of Suarez, who turned the ball into the net from point-blank range.
Barca, buoyed by their breakthrough, turned on the style to make it 2-1 with 16 minutes to play. The ball broke to Alves after a fine run from Messi, and his cross landed perfectly on the head of the unmarked Suarez, who powered the ball past Oblak's despairing dive.
Messi was denied a late penalty shout after tumbling under pressure in the crowded area as Barca were made to settle for a result which keeps the tie intriguingly poised ahead of the second leg.
Key Opta stats:
- Fernando Torres is the first player to score and get sent off in a UCL game since Yaya Toure in November 2014.
- Until Torres' goal, Diego Simeone's men had failed to find the net in their last 508 minutes of action in the Champions League knockout stages.
- Barcelona have only kept one clean sheet in 13 Champions League/European Cup games against Spanish opposition - against Real Madrid at the Bernabeu in April 2011.
- Luis Suarez is the top scorer in all competitions in the top five European Leagues (45 goals).