City's Bernardo Silva and Mateo Kovacic both misfired in the shootout to put Madrid in the driving seat
Suarez broke Valencia's resolve in the semi-final first leg by leaping to meet a cross from Lionel Messi in the 67th minute, scoring a 12th goal in 10 games in all competitions to leave Valencia needing to win by two goals in next Thursday's second leg at Mestalla.
Barca defender Sergi Roberto risked getting sent off but was only booked for a wild lunge on Andreas Pereira, who was unable to come out in the second half and had to be substituted.
Valencia were trounced 7-0 on their last Cup visit to Barcelona when Gary Neville was in charge and coach Marcelino opted for a cautious approach.
The visitors' best chances came from distance strikes at goal, with Martin Montoya and Francis Coquelin narrowly missing the target either side of halftime.