City's Bernardo Silva and Mateo Kovacic both misfired in the shootout to put Madrid in the driving seat
Suarez crashed in an emphatic overhead kick in the 25th minute after a surging run by Messi then teed up the Argentine to knock in the second three minutes later.
The Argentine also rattled the bar from outside the box early in the game and his brace took his tally to 29 goals in 30 games against Sevilla in all competitions. Messi also leads the Liga scoring charts with 27 strikes, while Suarez has 24.
"We took the game to them as much as possible, we had more of the ball, we scored three goals and had a lot of chances, which says a lot when you consider the quality of our opponents. Things changed in the second half but everything was under control."