Messi opened the scoring after two minutes and eight seconds, the quickest goal of his career, slipping the ball through Thibaut Courtois' legs from a tight angle.
The Argentine forward created the second with a blistering run from the halfway line which left three Chelsea defenders stumbling in his wake, before squaring the ball for Ousmane Dembele to rifle home his first goal for the club.
Marcos Alonso hit the post and Olivier Giroud was unfortunate not to be awarded a penalty for a tangle with Gerard Pique as Chelsea tried to launch a comeback, before Messi killed the tie with his 100th Champions League goal.
"To have Messi in the team is a luxury, he's such a good player that in any move he can create danger against you," Barcelona coach Ernesto Valverde said.
"Our attacking play goes through him and any move is improved by his touch."
Chelsea manager Antonio Conte said his side needed a perfect display to progress but they started disastrously at a packed Nou Camp when Luis Suarez teed up Messi, who squeezed a low strike through the legs of Courtois from a tight angle.
The visitors responded well with Giroud and Willian, who scored in the 1-1 first-leg draw at Stamford Bridge, particularly bright in attack.
But a mistake from Cesc Fabregas on the halfway line allowed Messi to break in again after 20 minutes and the number 10 left the Chelsea backline in tatters to set up the second goal, expertly finished by Dembele.
-Reuters