Sterling goal sees Liverpool past Olympiacos
Brendan Rodgers' men were buoyed to victory by a fifth-minute goal to Raheem Sterling, who benefitted from a deflected shot from strike partner Daniel Sturridge.
The Reds were good value for their win, although Olympiacos were inches away from earning a point, with Gevorg Ghazaryan denied by the crossbar early in the second half with Brad Jones beaten.
Liverpool's first match of the pre-season tournament in the US saw them start well, with a free-kick in the fourth minute in a dangerous area, before they took the lead a minute later.
Sturridge's attempt from distance deflected off an Olympiacos defender, and fell fortunately to Sterling, who tucked away past a helpless Balazs Megyeri.
The game began to open up later in the half, with Liverpool producing several fast counter-attacks, with new signing Lukas Markovic involved.
Olympiacos had appeals for a penalty waved away in the 35th minute, as Javier Saviola earned a yellow card for simulation inside the area.
Moments later, Markovic showed elements of his class as he danced into the Olympiacos area, only to guide his curling right-footed shot too high for the top-right corner target.
Rodgers swung the changes at half-time, taking out Markovic, fellow signing Emre Can and Jose Enrique, and bringing in Joe Allen, Jack Robinson and Coutinho.
However, it was Olympiacos who should have struck first on return to the Soldier Field playing arena, as Ghazaryan's shot was agonisingly denied by the woodwork.
Andreas Bouchalakis then had a gilt-edged chance to equalise, forcing a turnover deep in Liverpool's defensive third, however he could not control his final touch around goalkeeper Jones as the chance dissipated.