Bearne scored the only goal of the game after half an hour at Finch Farm to maintain Liverpool's title hopes.
The result means Barry Lewtas' side go level on points with Derby County at the top of the U18s Premier League North table.
Liverpool emerged the victors of a dramatic 4-3 encounter with their local rivals back in November and came through another end-to-end meeting on the winning side.
After a number of early sightings for Everton forwards Ellis Simms and Lewis Dobbin, Liverpool settled and began to assert control on the affair.
Elijah Dixon-Bonner tested Harry Tyrer's alertness in the Everton goal with a powerful long-range effort five minutes before the opener arrived.