Despite their comprehensive win at Anfield, Liverpool's chances of progressing in the Champions League were dealt a blow in Portugal as Porto claimed three points against Group A rivals Marseille.
A draw would have been ideal for the Reds and it was tied at 1-1 until Lisandro Lopez scored the winner in the 79th minute.
Ricardo Quaresma was the architect, crossing precisely for Lopez to head home emphatically.
Tarik Sektioui scored first, putting the hosts into a 27th-minute lead with a stunning left-footed strike from the edge of the penalty area before Marseille struck back two minutes into the second half.
Mamadou Niang was the scorer, nodding home Laurent Bonnart's cross.
Rosenborg increased their chances of progressing from Group B - where Avram Grant's Chelsea currently top the table - by overturning Valencia 2-0 at the Mestalla.
Former Tottenham and Wolves striker Steffen Iversen was the hero, hitting the target in each half, first with an 18-yard drive and then with a well-hit header.
Celtic's chances of qualification also received a fillip, with AC Milan seeing off Shakhtar Donetsk 3-0 away from home.
The Ukrainian side are the Scots' biggest rivals for a spot in the next stage but found no answer to second-half strikes from Filipo Inzaghi (66, 94) and Kaka (72).
In Group C, Lazio's 2-1 victory at home to Werder Bremen put them firmly into contention.
Tomasso Rocchi's brace proved clinical as he converted from the penalty spot after 57 minutes before doubling the lead 11 minutes later.
Ribas Diego hit a last-minute penalty of his own to produce a dramatic finale but it was not to be for the Greek side.
Table-topping Real Madrid, meanwhile, could not force anything better than a goalless draw at Olympiacos.