The Reds welcomed back several players who represented the Under-23s in last Friday's game against Southampton for a tough test against the table-topping Merseysiders, following the loss to Middlesbrough at the weekend.
It was always expected to be a battle with the Blues and so it proved, but Levitt kept his composure to score from the spot in the closing stages after Neil Ryan's side had been frustrated by the woodwork on three occasions.
It was no more than United deserved after dominating the second half against a Toffees side that was forced to play almost two-thirds of the game with only 10 players after defender Josh Hosie was sent off for two yellow-card offences - fouls on Largie Ramazani and Brandon Williams as they made tracks down the Reds' left-hand side.