Villas-Boas admits the Bluesa fans have become "anxious" after Chelsea were dumped out of the Carling Cup after a 2-0 defeat by Liverpool.
The pressure on Villas-Boas intensified as goals from Maxi Rodriguez and Martin Kelly subjected the Blues to their third home defeat in four games.
It was a feeble and toothless display from Chelsea, who face critical fixtures at St James' Park ... then at Stamford Bridge against Valencia and Manchester City over the next two weeks.
Andre Villas-Boas: "Away from home we've been more solid statistically and as a team. At home we just haven't been good enough.
"We need to get our fans behind us and get the emotion right. You can feel Stamford Bridge has become anxious.
"We therefore need support. We need atmosphere to get us past this period.
"We have big games where we need our fans behind us and I'm sure they'll do exactly that."
CHELSEA'S CURRENT FORM:
Chelsea 0 Liverpool 2
NOVEMBER 29: Carling Cup quarter-final at Stamford Bridge.
Chelsea 3 Wolves 0
NOVEMBER 26: Barclays Premier League at Stamford Bridge.
Bayer Leverkusen 2 Chelsea 1
NOVEMBER 23: Champions League Group E at the BayArena.
Chelsea 1 Liverpool 2
NOVEMBER 20: Barclays Premier League at Stamford Bridge.
Blackburn 0 Chelsea 1
NOVEMBER 5: Barclays Premier League at Ewood Park.
Genk 1 Chelsea 1
NOVEMBER 1: Champions League Group E at the KRC Genk Arena.
CHELSEA'S LAST GAME:-
Chelsea 0 Liverpool 2
CHELSEA: Turnbull, Bosingwa, Luiz, Alex, Bertrand, Lampard, Romeu, McEachran (Ramires 41), Malouda (Anelka 64), Torres, Lukaku (Mata 64).
Subs Not Used: Hilario, Ivanovic, Ferreira, Kalou.
LIVERPOOL: Reina, Kelly, Carragher, Coates, Jose Enrique, Henderson, Lucas (Adam 70), Spearing, Maxi (Skrtel 89), Bellamy (Kuyt 79), Carroll.
Subs Not Used: Doni, Suarez, Downing, Flanagan.
Att: 40, 511
Craig Bellamy steered Liverpool into the Carling Cup semi-finals to heap further pressure on under-fire Chelsea manager Andre Villas-Boas.
Bellamy's pinpoint delivery set up second-half goals for Maxi Rodriguez and Martin Kelly as the Reds took a step closer to winning their first silverware since 2006.
It completed an emotional return for Bellamy, who had been omitted from the squad for Sunday's 1-1 draw with Manchester City on compassionate grounds following the death of close friend Gary Speed.
The 32-year-old Wales winger provided the spark of creativity Liverpool needed to defeat toothless opposition who were prone to erratic defending.
The ineffectual Andy Carroll, who started at Stamford Bridge, missed a first-half penalty while Chelsea should have been awarded a clear spot-kick after just two minutes.
Villas-Boas will be disappointed by the poor defending that led to both goals and hastened Chelsea's third defeat in four matches, a sequence spanning just 10 days.
The Portuguese is now in a precarious position, with his future certain to attract fresh scrutiny as the Blues continue to flounder under his guidance.
Adding to Villas-Boas' woes was the performance of anonymous £50million striker Fernando Torres, who was a peripheral figure against his former club.
Kenny Dalglish has now never lost to Chelsea as Liverpool manager in 13 meetings, but that record would have been more seriously challenged had referee Phil Dowd not blundered in the third minute.
David Luiz had dashed into the area but just as he appeared to have rounded Sebastian Coates, he was caught by the Uruguayan's knee and tumbled over.
Dowd was unimpressed, however, and booked Luiz for diving.
Alex clattered into Bellamy in a feisty opening that soon settled down as both teams sought to pass their way into the final third.
Dowd did point to the penalty spot in the 21st minute and it was Liverpool who were the beneficiaries.
Alex blatantly handled the ball while under pressure from Carroll and Dowd hesitated before awarding the kick and booking the Chelsea defender.
Carroll, who had protested fiercely to Dowd, stepped up only to drill his effort straight at Ross Turnbull, allowing the Blues to wriggle off the hook.
Chelsea twice passed their way out of trouble in their own third of the pitch, but the same patterns were missing in the opposition's penalty area.
The limping Josh McEachran was replaced by Ramires in the 42nd minute, moments before Florent Malouda almost connected with a dangerous through-ball.
Dowd made another mistake by booking Ryan Bertrand for a nasty challenge on Jordan Henderson when it was Romelu Lukaku who caught the midfielder on the shin with his studs.
Lukaku then sent a header narrowly wide after being expertly teed up by Jose Bosingwa.
Bellamy almost released Carroll after dashing clear from his own half, but he overhit the pass and England marksman had no chance of connecting.
Lampard sent in a free-kick that caused panic in Liverpool's defence, with first the crossbar and then Coates coming to their rescue.
Luiz stopped Rodriguez in his tracks with a wonderful tackle as the former Argentina international charged into the area.
But Liverpool and Rodriguez could not be denied in the 58th minute when Liverpool surged into the lead.
Bellamy did the initial damage with a killer ball across goal - perhaps Luiz should have done more to prevent the pass - for Rodriguez to complete a simple tap-in.
Liverpool's large and noisy travelling contingent were back on their feet five minutes later as Liverpool grabbed a second.
Once more Bellamy supplied the ammunition - this time from a free-kick - and Kelly was given all the time in the world to nod home his first goal for the club.
Liverpool midfielder Lucas was stretchered off after going down in a tackle.
Torres was denied by the outstretched arms of keeper Pepe Reina, the Spain striker's most meaningful contribution to a match that had largely passed him by.
Bellamy left to a richly deserved standing ovation from visiting fans, who had earlier taunted Villas-Boas by chanting "you're getting sacked in the morning".