Atletico dominated the game and hit the woodwork twice in the first half before having two goals ruled out for offside but fell behind in the 76th minute when a bullet header from Ante Budimir crossed the line before Jan Oblak could punch it away.
Atletico hit back with an emphatic finish from substitute Renan Lodi in the 82nd minute while top scorer Suarez completed the comeback by firing home from inside the area in the 88th.
Diego Simeone's side nearly conceded an equaliser deep into added time though when Chimy Avila caught sight of goal but Oblak got across to parry before grabbing the ball with both hands.
As they had done in Wednesday's 2-1 win over Real Sociedad, Atletico came flying out of the blocks in search of a goal to put their minds at ease.
Atletico's first clear chance fell to Suarez but the Uruguayan scuffed his shot inside the area and Herrera saved well. Suarez then wasted an even more glaring effort, hitting the post from only a few metres out with Herrera beaten.
The visiting keeper made his fourth save in the first half to keep out a shot from Angel Correa while he was beaten by a long range strike from Saul Niguez but the shot kissed the outside of the post.
Atletico continued to search hungrily for an opening in the second half but the nerves seemed to be setting in and they could not keep their composure in front of goal.
They did manage to find the net when Stefan Savic knocked in a free kick but the goal was ruled out for offside while Carrasco also netted a few minutes later only to have his effort chalked off.
The title looked to be slipping from Atletico's fingers when Croatian forward Budimir beat Oblak, just as Real took the lead in Bilbao.
But Simeone's side responded perfectly with two goals in the space of six minutes.
"The team played a great game," Simeone said after the match. "We had a great first half where we created chances, we weren't clinical but controlled the game, then Budimir scored a good goal, and the team responded. When we had fewer chances we were more clinical."
Simeone added of Suarez's heroics: "Luis came here wanting to prove himself from the first day. It's true he's been far from goal but he's always been there working for the team. He has authority. Who better than him to win a game that we were able to sort out when it was getting away from us."