He scored his second - his 100th goal in the competition - in the second half.
"He knows in certain moments the stars have to come out," said Barcelona coach Ernesto Valverde. "We enjoy him. We're lucky enough to be seeing something that will go down in history."
Messi reached the landmark on his 123rd appearance in the Champions League. Cristiano Ronaldo has 117 goals in 152 games.
"I think that a fantastic player moved the final result," said Chelsea manager Antonio Conte. "In these two legs, he scored three goals and he made an assist for Dembele to score a goal."
Chelsea went into the last 16 second leg at the Camp Nou with the tie finely balanced at 1-1 following the first leg at Stamford Bridge. In west London, Chelsea were twice denied by the woodwork and the hoodoo continued in Catalonia.
"If you watch the game you can see the final result is unfair," Conte said. "Over the two legs we hit the post four times. This is very strange."
Barcelona form part of a three strong Spanish contingent in the quarter-finals. Roma and Juventus are Italy's representatives while Manchester City and Liverpool join the party from England.
Bayern Munich advanced on Wednesday night to the last eight following a 3-1 victory in Besiktas. The second leg in Istanbul was academic after Bayern romped to a 5-0 win in the first leg. Vagner Love at least salvaged some pride for theTurkish outfit with a goal in the second-half at the Vodafone Stadyumu.