The Reds required a victory over the Swiss champions to join Real Madrid in the Round of 16 but were unsuccessful in their pursuit of making it through on their return to Europe's top flight.
Fabian Frei made life difficult for Brendan Rodgers' men when firing home from 20 yards in the 25th minute before Lazar Markovic's sending off just after the hour-mark heaped more pressure on the Merseysiders.
Steven Gerrard attempted to lift his side by netting with an 81st-minute free-kick but the Reds were unable to find the winning goal needed to progress to the knockout phase of the competition.
Basel will join Real in the next stage after the reigning European champions comfortable defeated Bulgarian minnows Ludogorets Razgrad 4-0 at the Santiago Bernabeu.
Cristiano Ronaldo netted his 30th goal in all competitions this season when converting from the penalty spot in the 20th minute after Marcelinho had been sent off for handling the ball in the area.
Gareth Bale added a second seven minutes before the break prior to second-half goals from Alvaro Arbeloa and youngster Alvaro Medran - the latter opening his Champions League account - to give Carlo Ancelotti's side six wins from six group stage fixtures.
Lukas Podolski gained a rare start and did not disappoint when thundering home in just the third minute before a brace from Aaron Ramsey - the second a stunner from 30 yards - had the Gunners in complete control.
Wednesday night sees Manchester City needing to beat AS Roma in Italy to move through to the knockout phase as Barcelona host Paris Saint-Germain while Sporting Lisbon travel to Chelsea and Bayern Munich are at home to CSKA Moscow.
2014/15 UEFA Champions League - group stage scores
Olympiakos (GRE): 4 (David Fuster 22', Alejandro Dominguez 63', Kostas Mitroglou 87', Ibrahim Afellay 90') Malmo (SWE): 2 (Simon Kroon 59', Markus Rosenberg 81') Karaiskakis Stadium, Athens, Greece - 27,562
Real Madrid (SPN): 4 (Cristiano Ronaldo (p) 20', Gareth Bale 38', Alvaro Arbeloa 79', Alvaro Medran 88') Ludogorets Razgrad (BUL): 0 (red card: Marcelinho 19') Santiago Bernabeu, Madrid, Spain - 58,393
Borussia Dortmund (GER): 1 (Ciro Immobile 58') Anderlecht (BEL): 1 (Aleksandar Mitrovic 84') Westfalenstadion, Dortmund, Germany - 65,851 *Borussia Dortmund and Arsenal progress to Round of 16 *Anderlecht drop into Europa League
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