Koke handed the visitors a deserved early lead, but Barca turned the game thanks to two goals in eight minutes before the break through Messi and Suarez.
Atletico’s hopes of salvaging a point were then ruined as Filipe Luis and Diego Godin were sent-off for rash challenges either side of half-time.
Victory takes Barca three points clear of Atletico at the top of the table – having also played a game less – and seven points clear of third-placed Real Madrid, who host Espanyol on Sunday.
“It is clear it is an important day for us to pick up three points against our closest rivals, but there is a long way to go,” said Barca boss Luis Enrique.
“Every game presents its difficulties and you don’t always overcome them in the most effective or spectacular way.”
Meanwhile, Atletico coach Diego Simeone refused to chastise Luis and Godin for leaving his side short-handed against the European champions.
“I have nothing to reproach them for,” said the Argentine.
“Even with nine men we maintained our way of playing. In a game you can win or lose, but I always prefer to lose in this way.”
The clash between La Liga’s top two pitted Europe’s most feared attack against its strongest defence, but it was Atletico who started on the front foot.
Saul Niguez’s dipping effort was brilliantly turned over by Claudio Bravo after just two minutes.
However, another Niguez burst provided the opening goal eight minutes later when his cross from the right just evaded Antoine Griezmann and was slotted home at the far post by Koke.
Atletico could even have had a more commanding lead as Augusto Fernandez flashed a shot inches wide with Barca struggling to get a foothold in the game.
The European champions had been outplayed by Malaga and Athletic Bilbao in the first-half of their previous two games before reacting after the break, but they clicked into gear in the final 15 minutes of the first-half to turn the game on its head.
Suarez fired a warning shot that Jan Oblak got down well to save at his near post.
However, the Slovenian’s 483-minute run without conceding was ended when Jordi Alba showed great composure to pull a low cross into Messi’s path to smash home.
Atletico were then caught uncharacteristically flat-footed as Dani Alves’s long ball over the top picked out Suarez, who showed great strength to hold off compatriot Jose Maria Gimenez, before slotting between Oblak’s legs.
The game looked over as a contest a minute before the break when Luis was sent-off for high, studs-up, challenge on Messi inside the Barcelona half.
Yet, despite their numerical disadvantage, Atletico still started the second period the better.
Yannick Carrasco’s lung-bursting run was snuffed out at the last hurdle by Gerard Pique.
The Belgian then produced a fine cross for Griezmann, but his acrobatic effort from point-blank range was brilliantly saved by Bravo.
Atletico’s resistance was undone by their own stupidity again 25 minutes from time, though, as Godin lunged in on Suarez and was rightly shown a second yellow card.
Barca then rubbed salt into the visitors’ wounds by introducing former Atletico midfielder Arda Turan for Ivan Rakitic and the Turkish international was inches away from his first goal since swapping the Spanish capital for the Camp Nou when his shot flew just wide of the far post.
And a horrible day for Diego Simeone’s men was rounded off when Fernandez had to be stretchered from the field with what looked like a serious knee injury 15 minutes from time.