The London side host Spanish club Valencia next week in their final Group E fixture knowing defeat or a draw would knock them out of the competition at the group stage for the first time.
"It was not good today," Portuguese Villas-Boas, who replaced Carlo Ancelotti in the Chelsea hot seat in the summer, told reporters. "It's a pity for us because we had a good chance to progress at home but Liverpool were superior today.
"In terms of the Champions League it's life or death - this level won't be good enough. We have to up the tempo and play that game with the ultimate desire."
Villas-Boas said the next month would be a crucial period for the club who are currently outside the top four in the Premier League.
"We have to do the maximum with our fixtures. December fixtures include Newcastle, Man City and Tottenham but for us to progress we have to make the most of these games in December," he said. "Bearing in mind that we play these top teams it can put our challenge back on track.
"At the moment at home we just haven't been good enough. I think we need to get our fans behind us. You know that Stamford Bridge [fans] have become anxious about playing at home. We have two big games at home where we will need our fans behind us."
Second-half goals by Maxi Rodriguez and Martin Kelly, both created by Craig Bellamy, made it a good night for Liverpool manager Kenny Dalglish whose side have now won at Chelsea twice in 10 days following a 2-1 league win.
An injury to midfielder Lucas took some of the gloss off the night though, the Brazilian carried off on a stretcher with a knee injury.
"Lucas has damaged his knee, we don't know what it is," Dalglish said. "Rather than speculate we'll get a proper diagnosis tomorrow or the next day."
Dalglish praised the performance of Welshman Bellamy who was in tears before the match during a minute's silence for Wales manager Gary Speed who died on Sunday, aged 42.
"We know he's a fantastic footballer," said Dalglish. "To come back and play like that after the man he considered a mentor passed away. For the wee man it's great for him."
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