The holders should have won comfortably but their failure to take more than one of many chances meant that they had to endure an edgy conclusion before inflicting Anderlecht's 10th successive defeat in the competition.
Should Liverpool extend that run to 11 at Anfield in 12 days, and Real Betis lose to Chelsea the same night, or fail to win at Anfield on November 23, Liverpool's much-anticipated visit to Stamford Bridge on 6 December will be rendered academic.
"It was a difficult game because we created a lot of opportunities, but we never scored the second goal so they continued going forward," said Rafael Benitez, the Liverpool manager.
Cissé has appeared the last resort when Benitez has been selecting a centre-forward, but he drew deserved praise from his manager last night. "He has done a good job for us," Benitez said. "For a striker it is really important to score goals and now I will have more options."
One of those is Harry Kewell who came on, to a lukewarm reception from the otherwise vociferous travelling support, for the last 16 minutes. It was the Australian's first appearance since limping out of last year's final.
Like Arsenal in Prague the previous night, Liverpool appear able to leave their domestic inconsistencies behind them on the European stage. This particularly applies to Cissé, who has now scored seven times in continental competition this season.
That record, and Saturday's Premiership winner against Blackburn, earned him the centre-forward slot with Fernando Morientes and Peter Crouch left on the bench. With Steven Gerrard injured, Luis Garcia was asked to operate in the hole off Cissé and Mohamed Sissoko on the right flank of a four-man midfield.
But it was Anderlecht who made the initial running. In the second minute Christian Wilhelmsson, who troubled Liverpool all night, glided through their midfield and defence before releasing Bart Goor. It should have been Anderlecht's first goal of the competition but the veteran's shot struck Jose Reina's outstretched leg. Though Reina was drawn into action a few minutes later, holding Anthony Vanden Borre's fierce long-range drive, Anderlecht would not get a better opportunity.
Liverpool, though, were able to make chances at will for an hour. Which was fortunate because they spurned them just as readily. Garcia was the prime culprit. First he sidefooted over from 10 yards from John Arne Riise's slide-rule cross, then he pulled his shot wide after Cissé had unsettled the home defence. The Spaniard was still missing chances target deep in injury time.
This perhaps explained why Riise, the next time he skinned Vanden Borre, decided to cut in and shoot instead of cross. It proved a valid decision. He brought a fine save from Silvio Proto and, from the subsequent corner, Dietmar Hamann picked out Cissé, who thumped in a volley.
Liverpool failed to build on the advantage. Their best opportunity to do so fell to Djimi Traoré following another Hamann corner, but Proto smothered his shot. Anderlecht might have levelled when Mpo Mpenza nodded Wilhelmsson's cross into Goor's path. This time the big Belgian thrashed the ball over.
The interval came and went and so did the chances. Riise volleyed wide after Jamie Carragher released him over the top; Vanden Borre drew a smart low save from Reina.
As the home pressure mounted Xabi Alonso was booked for sliding through the back of Wilhelmsson. Liverpool were happier with the Swiss referee's next intervention. A Wilhelmsson cross hit Carragher's arm as he sought to block. Massimo Busacca, well-placed, waved the appeals away.
"Maybe he did not move his hand to the ball but he stopped it going somewhere," said Frank Vercauteren, the Anderlecht coach, adding, "the referee was not Champions' League quality."
Neither, though was Anderlecht's finishing, with another Vanden Borre miss confirming their third successive 1-0 defeat in this campaign.