Suarez scored twice as the holders celebrated qualifying for the last 16 of the Champions League with another stunning display on Tuesday.
Barcelona discovered they had qualified from Group E before they had kicked a ball at the Camp Nou, the 1-1 draw between BATE Borisov and Bayer Leverkusen in Belarus ensuring their passage.
Fresh from crushing eternal rivals Real Madrid 4-0 in La Liga action at the weekend, they then produced a performance that the Italians could not match to secure top spot in the section.
"We know that every player has to fulfil their expectations," Suarez said.
"We played a perfect game, very complete, and we didn't leave any space in midfield. In the first half we put the game away.
"Neymar, Messi and I are here for the good of the team, for the good of Barca. We don't think individually."
Messi, making his first return since injury, also notched up two goals.
"The people were waiting for Messi to return and we're all happy that he scored," acknowledged Suarez.
Messi said there was an evident synchronicity with his teammates.
"With Neymar and Suarez we get on really well. We know we have a responsibility but we also want to have a good time," the Argentinian said.
"I was really excited to have that feeling of coming back and playing again. I enjoyed watching the team, but playing is better."