Xavi's side needed extra-time to beat Intercity 4-3 in the round of 32 last week, and the Catalan outfit will now face another third-tier side in the shape of Ceuta for a spot in the quarter-finals.
Real Madrid have been handed a much tougher test, with Los Blancos taking on Villarreal in what is one of four all-La Liga ties in the Copa del Rey.
Indeed, Real Betis will take on Osasuna, Real Sociedad will face Mallorca and Athletic Bilbao will tackle Espanyol in the other ties involving top-tier sides.
There are also three matches involving La Liga teams against Segunda Division teams, with Atletico Madrid facing Levante, Valencia taking on Sporting Gijon and Sevilla drawn with Alaves.
Barcelona boss Xavi was asked about the clash with Ceuta during his press conference to preview Sunday night's La Liga contest with Atletico Madrid.
"We are focused on tomorrow's game, one of the most difficult outings in La Liga. We will try to be protagonists and have control of the ball. We want to be ourselves," he said.
"As for the cup, we've been lucky in the draw for the first time since I've been Barca coach, but the other day we proved against Intercity that we have to show that we're better on the pitch.
"The trip won't be easy, but I think we've been lucky with the draw."
Barcelona are the most successful side in the history of this competition, having lifted the trophy on 31 occasions, with their last triumph coming in 2020-21.
Athletic have the second-most wins in the competition with 23, while Real Madrid are third with 19, but Los Blancos have not won the tournament since 2013-14.
Meanwhile, the holders Real Betis, who beat Valencia on penalties in last season's final, are bidding to lift the trophy for the fourth time this season.
Copa del Rey last-16 draw:
Ceuta vs. Barcelona
Levante vs. Atletico Madrid
Sporting Gijon vs. Valencia
Alaves vs. Sevilla
Real Betis vs. Osasuna
Villarreal vs. Real Madrid
Real Sociedad vs. Mallorca
Athletic Bilbao vs. Espanyol
Bordeaux 
Dunkerque 
Arsenal 
Chelsea 
Leicester 
Southampton 
Lazio 
Juventus 
Nizhny Novgorod 
Lokomotiv Moscow 
Lorient 
Paris Saint Germain 
Monaco 
Lille 
Everton 
Liverpool 
Manchester Utd 
Sheffield Utd 
Marseille 
Nice 
Atalanta 
Fiorentina 
Ajax 
Excelsior 
Heerenveen 
PSV 
CSKA Moscow 
Spartak Moscow 
Udinese 
Roma 
Brighton 
Manchester City 
Go Ahead Eagles 
Feyenoord 
Montpellier 
Nantes 
Real Sociedad 
Real Madrid 
Dnipro-1 
Dynamo Kyiv 
West Ham 
Shakhtar Donetsk 
Inter 
Torino 
Bayern Munich 
Eintracht Frankfurt 
FC Augsburg 
Werder Bremen 
RB Leipzig 
Borussia Dortmund 
SC Freiburg 
VFL Wolfsburg 
Burnley 
Sampdoria 
Como 
AC Milan