The 21-year-old Cuenca will join Ajax on a five-month loan deal having not appeared for Barcelona this season after impressing last term.
"Cuenca is a welcome reinforcement as we still are active in three competitions and not always steady with our wingers," Ajax coach Frank de Boer said after the 4-0 Dutch Cup quarter-final win over Vitesse Arnhem.
Iceland striker Kolbeinn Sigthorsson marked his second appearance of the season by scoring twice in the game.
Ajax let slip a 2-0 lead when they lost 3-2 at Arnhem on Sunday but in Emmen, where the Dutch Cup tie was played because Vitesse's Gelredome stadium was unavailable, Frank de Boer's team grabbed a seventh-minute lead through Danny Hoesen.
The home team twice went close to equalising when keeper Jasper Cillessen denied Patrick van Aanholt and a header from Mike Havenaar was cleared off the line.
Lasse Schoene then extended Ajax's advantage after 59 minutes and Sigthorsson, who has suffered with injuries this season, made it 3-0 in the 81st when he followed up to convert the rebound after Schoene's penalty was saved.
Sigthorsson completed the rout by grabbing his second goal in the 86th minute.
"It was a strange day because until this morning I wasn't a part of the squad but then I came in and scored twice," he said. "It feels great after being sidelined for so long."
Holders PSV Eindhoven beat Feyenoord 2-1 on Wednesday while a day earlier PEC Zwolle and AZ Alkmaar eliminated Heracles Almelo and second-tier Den Bosch respectively.