Barcelona boss Enrique made two changes to the team that started Wednesday's 6-1 win over Sporting Gijon, with Sergi Roberto and Gerard Pique replacing Javier Mascherano and Denis Suarez. Rafinha, meanwhile, was preferred to Andres Iniesta in central midfield.
As for Celta, Andreu Fontas missed out through suspension, while Pione Sisto dropped to the bench as Theo Bongonda was brought in alongside John Guidetti and Iago Aspas in the final third. Sergio Alvarez also replaced Ruben Blanco between the sticks as Berizzo shuffled his pack.
It took until the 14th minute for either side to register a shot on target, and that came from Barcelona after Neymar and Messi exchanged little touches in the final third, but the latter's powerful attempt was kept out by Alvarez. Neymar then had a half-chance of his own after driving into the Celta box, only to chip wide of the far post.
Five minutes later, Luis Suarez and Messi were both allowed the freedom of the Celta box, but both attackers hit the post within the space of 10 seconds as Celta survived another dangerous moment.
The visitors then had a half-chance of their own in the 22nd minute when Umtiti's loose clearance fell to Facundo Roncaglia inside the Barcelona box, but the midfielder could only fire wide of Marc-Andre ter Stegen's post.
The open nature of the contest meant that a goal was bound to arrive, and it was Barcelona that made the breakthrough in the 24th minute when Messi dribbled to the edge of the Celta box before curling into the bottom corner.
The Catalan giants continued to press forward as they searched for a second goal, and Rafinha should have done better from inside the Celta box 10 minutes before the break, with the Brazilian firing a weak shot straight at Alvarez as Celta struggled to contain the Spanish champions.
Barcelona had their second five minutes before the interval, however, when Messi threaded a pass into Neymar and the Brazilian chipped the onrushing Alvarez to double the home side's advantage with a classy finish.
Roberto had a glorious chance to send Barcelona 3-0 ahead early in the second period after breaking into the Celta box, but Alvarez was out to make a smart save, before Neymar curled the ball wide of the post moments later.
Celta boss Eduardo Berizzo wasted no time in making a second-half change, and it was Denmark international Sisto that was called to replace Bongonda, who struggled to make an impact in the final third of the field.
Barcelona were full of confidence, however, and scored their third of the night just before the hour when Rakitic swept home after Rafinha had driven into the Celta box, creating space for the Croatian international.
Three then became four in the 61st minute when Umtiti converted a low Messi cross to register his first goal for the Catalan giants and leave Celta facing the possibility of more punishment in the final 30 minutes.
Indeed, another Barcelona goal was to arrive three minutes later, and it was Messi with his second as the Argentine drove down the right before cutting inside and wrong-footing Alvarez with a clever finish.
Iniesta and Denis Suarez were both introduced for the final 20 minutes as Sergio Busquets and Neymar were rested, but that did not interrupt the flow of traffic as Barcelona searched for more goals on home soil. Messi had a super chance to complete his hat-trick 15 minutes from time, but the Argentine looked for a pass into Luis Suarez rather than releasing a shot and Celta managed to clear their lines.
Aspas had a couple of half-chances for Celta in the final 10 minutes, but the former Liverpool attacker could not get his name on the scoresheet as Barcelona kept a much-needed clean sheet.
Next up for Barcelona is the second leg of their last-16 Champions League clash against Paris Saint-Germain on Wednesday night, with the Spanish champions needing to overturn a 4-0 defeat from the first leg to progress. Enrique's side will then return to La Liga action with a trip to Deportivo La Coruna next weekend.
As for Celta, Berizzo's side will host Krasnodar in the first leg of their last-16 Europa League contest on Thursday night, before returning to league action at home to Villarreal next weekend.