Messi scored both goals -- one with a header for only the fourth time in the competition -- as Barca beat Ajax 2-0 to equal Raul's record of 71 goals in the tournament.
"I am happy for Leo because of the work he does for the team. He is the number one, the best," said Barca midfielder Xavi Hernandez.
Bayern went through after a 2-0 victory at home to Roma, PSG saw off APOEL 1-0 while Porto beat Athletic Bilbao 2-0 with their goalscorers Colombian Jackson Martinez and Algerian Yacine Brahimi both taking their tallies to four in this season's competition.
However, Chelsea missed a chance to also reach the knockout stages as they had to come from behind to snatch a 1-1 draw with minnows Maribor in Slovenia.
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The 2012 winners, though, are in a far better state than English champions Manchester City whose mystifying failure to perform well in the competition continued as they had two men sent off in an embarrassing 2-1 home defeat to Russian outfit CSKA Moscow, who recorded their first ever win in England.
Their results rounded off a poor week for English clubs in the Champions League with Liverpool losing and Arsenal drawing 3-3 with Anderlecht, the Gunners having let a 3-0 lead slip in the final half hour on Tuesday.
Porto's fellow Portuguese side Sporting Lisbon are also in with a chance of reaching the next phase as they revived their campaign with a 4-2 home win over Schalke 04.