The Republic of Ireland international made a very encouraging debut for the Clarets. An early save when turning Smith's header around the upright was just what he needed. Also saved well from Bodvarsson in the first half. Very vocal.
Sean Dyche's side will now entertain the winners of the third round tie between holders Manchester City and AFC Bournemouth, which takes place on Thursday evening.
Burnley, who fielded three debutants including Bailey Peacock-Farrell, Bobby Thomas and Josh Benson, scored in either half to seal their progression in the competition.
Josh Brownhill opened his account for the club with a thunderbolt from 25 yards out before Matej Vydra netted with a cool finish deep into stoppage time.
Sign up to our daily Burnley Express Today newsletter
The i newsletter cut through the noise
Given an early test by Ferguson and Bodvarsson, but would go on to limit the duo's productivity down Millwall's left hand side. Played some nicely weighted balls into the channels and supplied the pass for Brownhill's winner.
Here are the ratings...
Kept former Rovers midfielder Connor Mahoney quiet throughout. Protected his pocket of the pitch extremely well and was a useful outlet in the final third. Whipped in a number of threatening crosses from different positions on the pitch.
Another positive display from another Burnley debutant. Given a real test on his bow when faced with the physicality of Smith, but dealt with the challenge admirably and with a mature head. Made some well-timed challenges around the penalty area.
The Burnley centre back continues to show a natural ability to perform at the top level. Led the defensive line well, showed good anticipation to cover his full back, made key blocks and was good in the air. Denied by Bialkowski with a second half header.
The Dutchman was handed the captain's armband for the Carabao Cup tie at The Den. Did a job on both sides of the pitch, preventing Romeo and Ferguson from advancing too far up the pitch. A big help for both full backs.
The Clarets winger suffered an indifferent first half, but came into his own after the half-hour mark. Made a couple of outstanding passes into the inside channels, with Taylor the benefactor, and whipped in a catalogue of teasing crosses.
What can we say about this man that hasn't already been said. Outstanding again, with and without the ball. Always in the thick of the action, piecing play together and breaking it up. His first goal for the club, from 30 yards out, was an absolute peach.
A worthy partner in the middle of the park for Brownhill. Added a protective layer for the Clarets defence and picked up some good pockets of space inbetween the lines to influence the game. Carried the ball well and calm on the ball.
The striker didn't get the opportunity to get into the game having sustained an injury around the quarter-of-an-hour mark. His only contribution - which led to his knock - was a header over the top from Brownhill's set-piece.
Worked so hard throughout the tie, showing good strength to hold up the ball in the final third while he worked a number of openings with his movement off the shoulder. Twice denied in the second half, but took his goal extremely well to seal victory.
Another industrious display from the striker, working across the front to hold up the ball and link play while also dropping off to collect the loose balls. Always alert to everything around him and picked out Vydra for Burnley's second.
Replaced goalscorer Brownhill with just 20 minutes of the tie remaining, but still made an impact. Covered plenty of ground and infiltrated the Lions backline with a number of exquisite passes.