Harris' side go out on penalties
The Lions, featuring one overage player in Josh Wright, started on the front foot and were rewarded for their fast start when forward Aiden O'Brien gave the visitors a deserved lead in the sixth minute; the Irishman receiving the ball in the box before turning and firing past Swansea keeper Gregor Zabaret.
That's how it stayed until the 66th minute when the home side replied through Liam Shephard, the youngster slotting past Lions keeper Denzil Gerrar from Henry Jones' free-kick.
With neither side able to find a winner the tie headed to extra-time, but with both goalkeepers on top, penalties would ensure a winner.
Midfielder Wright made no mistake from 12-yards to set Millwall rolling, but misses from Charlie Penny and then Tom Bender proved crucial as Swansea converted their first four spot-kicks to secure a 4-2 win on penalties.
The Swans will now face either Plymouth Argyle or Watford in the Third Round of the competition.