The hosts―who left top scorer Luis Suarez on the bench for all but the last 14 minutes―were out of sorts for much of the first half, with frustration the main emotion emerging from a capacity Anfield crowd who were expecting a comfortable victory.
The breakthrough was finally achieved when Iago Aspas calmly stroked home his first competitive goal for the Reds after being found in the penalty area by Raheem Sterling nine minutes into the second half.
Stand-in home goalkeeper Brad Jones made a fine save from Michael Petrasso to keep the scoreline at 1-0, and with Suarez now on the pitch, the Reds grabbed a second when Sterling's wayward drive was turned into the opposing net by defender James Tarkowski.
There was time for a romantic FA Cup moment as Oldham's Anton Rodgers came off the bench to face the team managed by his father Brendan, but ultimately it was to be Liverpool's day despite the brave efforts of their visitors.
Here are six lessons learned from the game: