Atletico went ahead in the 10th minute through Koke but Messi equalised for Barca following a fine team move on the half hour.
Suarez struck his 31st goal in as many games after 38 minutes and a wild challenge from Filipe Luis on Messi earned the Atletico defender a red card before halftime.
Diego Godin was also sent off after receiving a second yellow card for a mistimed tackle on Suarez midway through the second half.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"We knew it would be a difficult game and we are happy with three points against a direct opponent," keeper Claudio Bravo told reporters.
"There is still a long way to go in the championship. We are going well at the moment in all the tournaments and it is good to get a win even though we were not at our best."
Barcelona moved on to 51 points, three ahead of Atletico, and have a game in hand.
Atletico coach Diego Simeone opted to play five in midfield with Antoine Griezmann the lone striker.
It meant they were able to outnumber Barcelona in the middle of the pitch early on and they threatened immediately with Bravo stretching to block a Saul Niguez strike.
Koke was then given space to stroke the ball home in the centre of the area from a Saul cross.
Gradually Barca began to impose themselves, with Suarez having a shot blocked by keeper Jan Oblak after a good passing move and they again probed the defence before Jordi Alba picked out Messi for the equaliser.
Atletico were sloppy defensively as Suarez latched onto a simple ball over the top by Alves and beat Oblak too easily from a tight angle.
The visitors' chance of a comeback were hit by a knee high studs up challenge by Filipe Luis on Messi that saw him dismissed.
Still Atletico pushed forward after the break and Griezmann was unlucky with a shot that came back off Bravo, who knew little about it, before heads began to drop after Godin's dismissal.
A second half Bruno Soriano penalty gave Villarreal a 1-0 victory over Granada that put them level on 44 points with third-placed Real Madrid, who face Espanyol on Sunday.
Jean-Sylvain Babin pulled back Roberto Soldado in the area and Soriano sent keeper Andres Fernandez the wrong way with his spot kick after 54 minutes.
Elsewhere, an Inaki Williams header gave Athletic Bilbao a 1-0 victory at Getafe, Xabi Prieto and Inigo Martinez were on target for Real Sociedad in a 2-1 win over Real Betis, and Eibar fell 2-1 at home to Malaga.
(Reporting by Tim Hanlon; Editing by Justin Palmer/John O'Brien)
