Following a 1-1 draw in the first leg, the visitors fell behind after just 128 seconds as Lionel Messi shot through the legs of Thibaut Courtois from a tight angle.
Messi then set up Ousmane Dembele to hammer home the second after Cesc Fabregas lost possession.
Marcos Alonso hit the post for Chelsea with a free-kick before Messi’s 100th Champions League goal – after a mistake by Cesar Azpilicueta – took the hosts out of sight.
Alonso was then denied a penalty after a challenge by Gerard Pique as Barcelona joined La Liga rivals Real Madrid and Sevilla in Friday’s quarter-final draw.
Chelsea, meanwhile, join fellow Premier League clubs Manchester United and Tottenham in exiting in the first knockout round.