Barca have lifted the trophy in each of the last four campaigns and have been present in all but one of the last eight finals, but they have ground to make up in the second leg of the last-16 tie at Camp Nou next week.
It took time for the away side's defence to find their feet, and Levante had a brilliant chance to double their lead in the seventh minute when Emmanuel Boateng drove into the penalty box before looking to pick out the bottom corner, but Cillessen was down to make a smart save.
Boateng had an even better chance to register in the 13th minute when he met a low cross from Coke, but a combination of Murillo and Cillessen managed to keep the ball out of the back of the net.
Levante did have their second in the 18th minute, however, when Boateng's clever pass found Borja Mayoral and the on-loan Real Madrid forward managed to squeeze his low effort through Murillo's legs and into the back of the net.
Levante goalkeeper Aitor Fernandez made two big saves in the spell that followed, first keeping out a strike from Philippe Coutinho before excellently denying Malcom, who had been sent through on goal.
Fernandez then denied Carles Alena from distance as the Spanish champions continued to create openings, although the door was open for Levante down the other end of the field.
Mayoral came close to scoring his second of the match late in the first period when the 21-year-old pounced on a rebound inside the Barcelona box before firing towards goal, but Cillessen made a smart stop.
The on-loan Los Blancos attacker then fired another attempt wide of the post moments later, before Malcom just failed to meet an Ousmane Dembele cross down the other end as Levante led 2-0 at the interval.
Sergi Roberto replaced Miranda at the interval as Barcelona boss Ernesto Valverde looked for a route back into the match, but Levante continued to cause problems and Cillessen had to keep out a strike from Rochina early in the second period.
Coutinho had Barcelona's first sight of goal in the second half, but the Brazilian's 52nd-minute effort was always rising over the home side's crossbar.
Cillessen then had to keep out another effort from the lively Mayoral moments later, before Moses Simon tested Barcelona's second-choice goalkeeper from distance.
Levante had another golden opportunity in the 64th minute when Jose Luis Morales was found by Mayoral inside the Barcelona box, but the substitute saw his effort saved over the crossbar by Cillessen.
Denis Suarez - strongly linked with a move to Arsenal - came off the bench in the 67th minute as a replacement for Malcom, who struggled to make his mark on the night.
Barcelona were having a lot more of the ball as the second period approached its latter stages, and the visitors had a penalty in the 84th minute when Denis Suarez was brought to the ground by Coke.
Coutinho stepped forward to make it 2-1 in the 85th minute and that was how it finished in Valencia to leave the tie delicately poised ahead of the return at Camp Nou next week.
LEVANTE (3-5-2): Aitor; Cabaco, Postigo, Chema (Jason 69'); Coke, Rochina (Doukoure 75'), Prcic, Campana, Simon; Mayoral, Boateng (Morales 62')
BARCELONA (4-3-3): Cillessen; Semedo, Murillo, Chumi (Lenglet 59'), Miranda (Roberto 46'); Busquets, Vidal, Alena; Coutinho, Dembele, Malcom (D Suarez 67')