Liverpool won the opening game of their pre-season warm-up competition on Sunday night – but they made hard work of overcoming Olympiakos 1-0 in Chicago.
Although Raheem Sterling scored after only five minutes in their International Champions Cup opener, Liverpool could not build on the lead and the Greeks narrowly missed out on a second-half equaliser.
Brendan Rodgers's pre-season preparation was disrupted when £27m signing Adam Lallana injured knee ligaments in training and will be out for four to six weeks.
But the Liverpool boss, who will be hoping his team can go one better than last season and win the Premiership title for the first time in 25 years, could draw some encouragement from the performance of another new face, Lukas Markovic.
Signed for £20m from Benfica, the 20-year-old Serb showed some good touches in the win at Soldier Field and came close with a solo effort in the first half before making way for fresh legs after 45 minutes.
Liverpool started the match strongly and when Daniel Sturridge's long range effort hit a defender, his England colleague Sterling was there to pick up the rebound.
The Reds were the better team but at the start of the second half, Olympiakos midfielder Gevorg Ghazaryan was denied by the crossbar with Brad Jones beaten.
Andreas Bouchalakis then had a great chance to score with 10 minutes to go when a mistake left him clear on goal. But his touch was too heavy when he tried to go around Jones and the opening was wasted.
Liverpool's next game is in New York on Wednesday when they face Manchester City. The English champions beat Milan 5-1 in their opening ICC game on Sunday in Pittsburgh.