The Foxes fell behind in the tie at Brunton Park, but hit back spectacularly to book their place in round three.
Chris Wood stole the headlines with an impressive hat-trick, while Lloyd Dyer and Anthony Knockaert were also on target as the Championship side strolled through.
On the cup success, Shakespeare told the club's official website: "We went a goal behind and that got the crowd going, but after that we managed the ball well and opened them up on many occasions. We probably should have scored more goals.
"We're at our best when we pass with precision and we were very pleased with the end result.
"Tonight was pleasing for Chris, he deserved his goals. He's a team player who is very appreciative of the players around him and his finishing tonight was clinical.
"Lloyd Dyer scored a great goal on the counter-attack too and his pace will always cause problems for defenders.
"The players have worked hard tonight and it's a long trip home now, so they'll get their feet up on the coach, have a day off tomorrow and come back in ready to prepare for Charlton on Saturday."