The Blues raced into a two-goal lead through Conor Gallagher and Cole Palmer before United also scored twice, through Alejandro Garnacho and Bruno Fernandes, to level the game at 2-2 by half-time.
Antony's superb outside-of-the-foot pass set up Garnacho to make it 3-2, with the Reds heading into the final stages looking set to register a famous comeback win.
With eight minutes of injury time already elapsed, Chelsea were awarded a penalty with Diogo Dalot having adjudged to have fouled Noni Madueke. Cole Palmer stepped up to make it 3-3, before the Three Lions international smashed home the winner with almost the last kick of the game.