Atletico scored a potentially valuable away goal when Fernando Torres blasted in a clinical first-time shot.
But the game turned when Torres was sent off before half-time for two needless bookings inside seven minutes.
Holders Barca upped the tempo after the break, Suarez prodding in the equaliser then powerfully heading in the winner.
However, the Uruguayan had been fortunate not to be shown a first-half red card for kicking out at Atletico right-back Juanfran.
The teams will meet again in the second leg at the Vicente Calderon next Wednesday.