Sevilla, who began the night leading the way in Group E, were stunned 5-1 by Spartak Moscow.
Spartak scored four times in the second half - through Lorenzo Melgarejo, Denis Glushakov, Luiz Adriano and Quincy Promes - to join Liverpool on five points at the halfway stage, with Sevilla one behind on four.
Jurgen Klopp's side were impressive 7-0 winners away to Maribor.
Manchester City's 100 per cent start to Group F continued as they defeated Napoli 2-1 at the Etihad Stadium.
Elsewhere in City's group, Bernard scored twice as a Shakhtar Donetsk side that finished with 10 men overcame Feyenoord 2-1.
Tottenham held Champions League holders Real Madrid to a 1-1 draw in Spain to remain undefeated in Group H. The pair are now six points clear of Borussia Dortmund and Apoel Nicosia, who played out a 1-1 draw in Cyprus.
Two errors from Dortmund goalkeeper Roman Burki led to Mickael Pote opening the scoring for the hosts, with Sokratis Papastathopoulos' equaliser at least ensuring the visitors returned to Germany with one point.
Monaco's dreadful campaign continued as last year's semi-finalists were beaten at home by Besiktas.
An opening goal from ex-Manchester United and Chelsea striker Falcao was cancelled out by Cenk Tosun, who then scored again in the second half to earn the Turkish club a 2-1 success.
Monaco are now bottom of Group G - with just a solitary point from three games - as RB Leipzig defeated Porto 3-2, with all five goals coming in the first half.