City's Bernardo Silva and Mateo Kovacic both misfired in the shootout to put Madrid in the driving seat
Suarez put Barca in front at a packed and rocking Camp Nou in the 29th minute after having an earlier effort ruled out for offside before the Uruguayan set up Messi on his landmark night four minutes later.
Dortmund's England forward Jadon Sancho scored an impressive consolation strike for the Germans in the 77th minute and then forced Barca keeper Marc-Andre ter Stegen to turn a shot on to the crossbar.
Ernesto Valverde's side top the group with 11 points from five games, Inter Milan are second on seven while Dortmund are third, also on seven, with Slavia bottom with two.