The Reds had to wait until first-half stoppage time to score against the group stage debutants, as Dirk Kuyt fired in a rebound to earn the three points.
European giants Inter Milan and Barcelona had to settle for a point apiece after neither side could find a breakthrough in the San Siro.
Holders Barca fielded former Inter hitman Zlatan Ibrahimovic, while ex-Barca striker Samuel Eto'o played for Inter, but neither could manage a shot on target.
Dynamo Kiev took full advantage to go top of the group after coming from behind to beat Champions League new boys Rubin Kazan from Russia.
Alejandro Dominguez curled in a free-kick to put the visitors ahead, but Ayila Yussuf headed in a leveller and Gerson Magrao's rifled finish and Oleg Gusev's close-range goal earned Dynamo the points.
Spanish side Sevilla picked up three points at the Ramon Sanchez Pizjuan Stadium as they saw off another Champions League debutant in the shape of Romania's Unirea Urziceni, who are managed by former Chelsea defender Dan Petrescu.
Sevilla had to wait until first-half stoppage time for Luis Fabiano to lash into the net from 12 yards, before the Brazilian headed on a corner to Renato, who bundled in from close range to make it 2-0.
The hosts took the lead when Pavel Pogrebnyak converted Cacau's pull-back, before Madjid Bougherra struck a fine second-half leveller to secure a point for the Gers.
On the day they announced Brazilian legend Zico as their new manager, Greek side Olympiakos secured a 1-0 victory at home against Dutch debutants AZ Alkmaar.
Vassilis Torosidis headed home Jaouad Zairi's cross with 11 minutes remaining as Zico watched his new team from the stands in Athens.
Arsenal produced a fabulous comeback to take an early lead in the the group as they fought back from 2-0 down to win 3-2 at Standard Liege.
Eliaquim Mangala shot home and Milan Jovanovic struck a penalty inside the first five minutes for the Belgians, but Nicklas Bendtner slid home, Thomas Vermaelen scrambled in and then Eduardo pounced for a dramatic winner.
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