Aside from a Marc-Andre ter Stergen save to keep out Shaquell Moore in a one-on-one opportunity, Barca eased to victory, extending their run to 26 matches unbeaten in all competitions.
The three stars may not be given much time to settle back into their familiar surroundings wiThernesto Valverde's side back in action on Thursday in their Copa del Ray clash against Celta Vigo.
No Levante player has scored in La Liga since November 19 and although the visitors made a bright start at a wet and half-empty Camp Nou, Barca opened their 2018 league account through talisman Messi after just 12 minutes.
Until the deal for Coutinho, Dembele was the club's most expensive signing for a transfer fee of 105 million euros that could rise to 147 million euros.More news: Trump should kill the Iran nuclear deal, for the dissidents and protesters
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I wanted to continue all my life here, I always said that I wanted to stay here for all my career and finish at Barcelona, it was my dream.
The second goal arrived seven minutes before the interval, as Suarez controlled Sergi Roberto's fine ball into the box before firing home. The striker controlled the ball with his left foot before sweeping home with his right for his 11th goal of the season and his sixth in a five-round scoring streak.
Messi was snapped with his two sons and wife Antonella Roccuzzo, who donned a bright pink Playboy jumper, as they arrived at the Barcelona-El Prat Airport.
The three points cancelled out second-placed Atletico Madrid's win over Getafe on Saturday to restore a nine-point advantage. "We need to have more confidence in our chances, and recently we don't have the confidence to play 90 minutes well".