The 18-year-old midfielder moved on to an Alan Smith pass and finished with aplomb into the bottom corner as United escaped a lottery of penalties by just two minutes. Sir Alex Ferguson was quick to pay tribute to Dario Gradi's battling side, who were arguably better over the 120 minutes.
Ferguson fielded a mixture of fringe players and youngsters as well as those returning from injury like Gabriel Heinze and Smith, for whom it was his first start since breaking his leg early in the year, but it was Crewe who called the shots early on with Julian Baudet rifling a 25-yard free-kick into keeper Tomasz Kuszczak's arms.
The Poalnd international did well to parry away Neil Cox's header which was bound for the far corner of the net from Andy Taylor's long free-kick and Crewe thought they had gone in front in the 16th minute when Isaac Osbourne saw his shot pushed away by the keeper, with Luke Varney finishing off from close range only to turn away from his celebrations when he spotted the offside flag raised against him.
It was United who took the lead rather against the run of play at the 26th minute when Ole Gunnar Solskjaer connected with a first-time shot into the bottom corner after Richie Jones had pulled the ball back from the right wing.
Home keeper Ben Williams was glad to get some assistance from his crossbar when he tipped Kieran Richardson's volley moments later over the top.
But Crewe were still in the game and Varney was a particular handful for the United defence, going close with a swirling shot after Jon Otsemobor and Nicky Maynard had combined.
The Premiership side took a grip of the game after the break without creating much and Crewe were still able to enjoy lots of possession and they secured a deserved equaliser in the 73rd minute, when Varney latched on to Maynard's pass and drove a low right-footed shot past Kuszczak at the near post.
But Richardson should have restored United's lead in extra-time but he stroked the ball on to the foot of the post after being put clean through by Solskjaer.
With time ebbing away Crewe fancied their chances in the penalty shoot-out but it was not to be as Lee struck in the 118th minute.