The opening 10 minutes were fiercely contested, with both teams having efforts on goal.
It was the visitors who took the lead on 15 minutes through Burrow, who jumped highest at the back post to convert a header.
Three minutes later, Lincoln had an opportunity to make it 2-0, but captain Alan Power fired wide after being cleverly picked out in the penalty area.
Barnet enjoyed a positive five minutes inside the Lincoln final third, but the away side doubled their lead on 22 minutes when Bencharif headed powerfully past the goalkeeper from close range.
In what was turning out to be a very open game, Barnet almost halved the deficit through John Akinde, but they were grateful to goalkeeper Graham Stack at the other end after he kept out a powerful effort from Bencherif.
Barnet had the better of the final 15 minutes of the half, but Lincoln held on for a commanding half-time advantage as Lee Cook had saw his first-time strike well saved.