Orient boss Russell Slade has come under increasing pressure in recent weeks, despite attracting the attention of Cardiff City, but he could be facing more uncertainty after seeing his team lose out on Tuesday evening.
It took the Blades less than two minutes to open the scoring at Brisbane Road as Higdon jumped above the defence to power home a header from eight yards.
Jamie Murphy fired a shot straight at the goalkeeper for United after cutting in from the left, while David Mooney's 35-yard attempt was never threatening the goal as the Blades held on for a half-time advantage.
Chances were few and far between at the beginning of the second half, but Orient slowly found their stride and struck the post from a 25-yard drive from Elliott Omozusi.
Niall McGinn failed to take advantage of a decent opening up the other end for United, and that invited pressure on his side as Orient increased the tempo in the closing stages.
Omozusi stung the palms of Mark Howard and McAnuff sent a shot wide from the edge of the penalty area, but Orient couldn't break through as Nigel Clough's side held on to book their place in the next round.