Fernando Torres sent Atletico into a 25th-minute lead on the night, but the Spaniard was then given his marching orders 10 minutes before the break after picking up two bookings in the space of six minutes.
Barcelona made their one-man advantage count in the second period as Suarez netted twice in 11 minutes to ensure that the reigning champions would take an advantage into the return at the Vicente Calderon.
Lionel Messi, Suarez and Neymar all started in the final third, while Dani Alves and Jordi Alba were given the nod in the full-back positions despite some criticism over their performance against Los Blancos.
Atletico, meanwhile, were boosted by the return of influential centre-back Diego Godin, who had been on the sidelines with a thigh problem since the middle of last month. The Uruguay international started alongside Lucas Hernandez in central defence, while Gabi skippered the team from the middle of the park.
There was also a start in the final third for Fernando Torres, who entered the match having netted ten career goals against Barcelona. Joining the Spaniard in attack was Atletico's top scorer Antoine Griezmann and fit-again Yannick Ferreira Carrasco.
The first attempt of the match came in the fifth minute when Neymar found Messi, who decided to release a strike from just outside the box, but the Argentine's low effort was always wide of the post.
Messi had another half-chance in the 13th minute after a cross from Alba found the attacker just inside the Atletico box, but the Ballon d'Or winner uncharacteristically sent a left-footed volley wide of the post.
The first serious opportunity arrived in the 19th minute when Alves found the head of Neymar with a cross from the right, but the Brazilian could only sent his effort over Jan Oblak's goal as Atletico remained unmoved at the back.
Against the run of play, it was Atletico that made the breakthrough in the 25th minute when Koke slid a pass in to Fernando Torres and the Spaniard smashed the ball home from close range after Barcelona's defence had parted.
Atletico continued to threaten as the first period developed and Barcelona stopper Ter Stegen had to be at his best to push a low Griezmann strike wide of the post in the 32nd minute of action.
The first leg swung in Barcelona's favour 10 minutes before the interval, however, when Fernando Torres, having opened the scoring, picked up a second yellow card following a late challenge on Sergio Busquets. On the touchline, Atletico boss Diego Simeone made his feelings known, but it was a clumsy challenge from the 32-year-old.
Barcelona did not actually have their first shot on target until the 41st minute and that was a tame effort from Javier Mascherano, which Oblak gathered into his chest. Neymar then tried his luck from distance in first-half stoppage time, but the attacker's effort sailed over as ten-man Atletico took a one-goal lead into the interval.
An over-head kick from Messi, meanwhile, just missed Oblak's post. Neymar then struck one towards the top corner after a slick move, but the attacker's effort smashed the crossbar during a pulsating few moments at Camp Nou.
Neymar again came close in the 56th minute after meeting a cross from Ivan Rakitic, but the Brazil captain headed straight into the arms of Oblak, who was then forced to get down and keep out a stinging Messi strike.
Atletico allowed Neymar to cut inside in the 62nd minute and the attacker looked to pick out the far corner, but his effort just missed the post as the visitors somehow clung onto their one-goal advantage.
Less than one minute later, however, Barcelona were level when Suarez, who had a quiet opening hour, turned a low Alba strike into the back of the net from close range to make it 44 goals for the season.
Griezmann sent one over the crossbar in the 71st minute as Atletico looked to cause problems in the final third, but the visitors spent the entirety of the second half on the back foot at Camp Nou.
A second goal seemed likely for Barcelona as they continued to press forward, and that arrived in the 74th minute when Suarez played a one-two with Alves before thumping a header past Oblak from close range.
The pace of the match slowed in the final 15 minutes as both teams seemingly settled for a 2-1 result and that was how it finished in Barcelona following a pulsating first leg of quarter-final action. Barcelona have now won each of the last seven clashes between the two teams in all competitions.