Fernando Morientes, plundering his 27th goal in the competition, set Anderlecht on the way to an 11th successive group defeat, with goals by Luis Garcia and Djibril Cissé compounding Belgian woes.
Chelsea's failure to end Real Betis's hopes means that the European champions must wait a little longer to qualify, but Liverpool could take satisfaction from a job completed by Cissé's shot with two minutes remaining. Anderlecht ended disgraced as well as defeated, Serbia's Nenad Jestrovic being sent off five minutes after coming on, for racist abuse of Mohamed Sissoko with the referee within earshot.
Quite apart from the question of reaching the knock-out stage, Liverpool needed a performance that built on Saturday's win over West Ham and put the failures at Fulham and Crystal Palace firmly behind them. Rafael Benitez made one change, restoring Peter Crouch for Cissé, and the striker had a half-chance in the opening minute but could not apply sufficient purchase to his header from Steve Finnan's cross.
Anderlecht's followers demanded a penalty when Akin Serhat fell like a wounded gazelle as John Arne Riise tackled him in the 11th minute. Kim Milton Nielsen, who has rather taken over from Anders Frisk as one of the European game's more self-regarding referees, rightly refused the Turk and could easily have penalised him for diving.
Moments after Sissoko was booked for cutting down Christian Wilhelmsson, Crouch had another opportunity to break his duck for Liverpool midway through the first half. Xabi Alonso sent Steven Gerrard scampering clear, his low cross being turned wide by Crouch at the near post.
The low-key atmosphere and aimless nature of the first half suited Anderlecht, who must have expected a tougher time at the home of the holders. In a bid to stir up his team, Gerrard suddenly materialised on the left flank. The captain's action had the desired effect, though not before another miss from Crouch. Finnan's 33rd-minute centre was met by Morientes, only for Silvio Proto to parry and the England striker to screw the rebound wide.
But within 60 seconds, Gerrard's sweeping cross-field pass from the left saw Morientes chest the ball down and dispatch an angled volley past Proto from just inside the area with his old aplomb.
Knowing that a second goal would probably foster a grim fatalism in the Anderlecht ranks, Liverpool pressed hard after the interval. Jamie Carragher - or Carradona, as the T-shirts on sale near the Shankly Gates proclaim - had a free header from Gerrard's 48th-minute corner but sent it tamely to a grateful Proto. Morientes soon gave way to Bolo Zenden, allowing Benitez to bring Garcia in closer to Crouch. In the programme, the Liverpool manager had written of the Spaniard benefiting from the space that the signing from Southampton created, although he was surely hoping that Garcia might conjure a breakthrough for Crouch.
When one of the new pairing did double Liverpool's advantage in the 61st minute - with a classic centre-forward's header - almost inevitably it was the smaller man. Alonso fed Finnan on the right, from where his cross was met by a flick of Garcia's forehead that propelled the ball across goal into the far corner beyond the diving goalkeeper.
