Messi, who also created Barcelona's third goal for Jordi Alba, set his team on its way to a 6-1 aggregate victory with his 21st and 22nd goals in all competitions this season.
Alba had provided the assists for Messi's strikes in the 13th and 15th minutes, and the Barcelona star returned the favor by setting up the left back to score in the 28th.
Luis Suarez made it 4-0 before halftime, while Messi was replaced by Ousmane Dembele before the hour mark. Ivan Rakitic headed in a corner kick by Dembele in the final minutes to complete the rout.
New signing Philippe Coutinho watched from the Camp Nou stands following his club-record transfer from Liverpool earlier this week. The Brazil midfielder will make his debut after he recovers from a right leg muscle injury.
Barcelona coach Ernesto Valverde took no chances after the 1-1 draw in the first leg, starting Messi in his first appearance in the competition this season after resting the Argentine forward for the first three cup games.
Messi quickly quashed any chance of Celta springing an upset.
Barcelona's all-time scoring leader swept Alba's cross past Sergio Alvarez despite the goalkeeper getting a hand to the ball, and he followed up by firing home after another exchange of passes with Alba.
Messi then turned provider, lobbing a perfect pass behind Celta's back line to meet the run by Alba, who chipped Alvarez to score.
"The connection between Messi and Alba is nothing new, it has been working for years," Valverde said. "They look for each other and are happy to do so, because it is not just Alba assisting Messi. Messi also creates goals for Alba."
Suarez netted in the 31st after pouncing on a poor pass by Pione Sisto, before Rakitic nodded home at the end.
Barcelona extended their unbeaten run this season to 28 straight matches, dating back to its losses to Real Madrid in both legs of the Spanish Super Cup in August.
Celta had drawn 2-2 at Barcelona in La Liga last month and had gone into Thursday's match after holding Madrid to a 2-2 draw in the league last weekend.
(With AP Inputs)