Ryan Babel gave put the Reds ahead with a cool finish in first-half injury-time before a hat-trick from Steven Gerrard and a deflected Sami Hyypia effort gave the scoreline a one-sided look.
Gerrard hoped the thrashing would refocus attention on to the pitch after this week's controversies.
"It's not just about the hat-trick, the win is the most important thing," he told Sky Sports 2. "It was important the headlines tomorrow were about football. Every time we've put the TV on we see what's happening off the pitch.
"It's not about the manager, its about the team and progressing into the fourth round. We need to keep winning so the headlines are about the football."
Jamie Carragher was captain for the night as he made his 500th appearance for the Merseysiders.
Asked whether he had ever expected to make the landmark, he added: "I never thought that. Even now I don't look too far ahead. You never know what's around the corner."
Luton manager Kevin Blackwell admitted his patched-up side were outclassed but he was proud of the way his players performed against a team he feels could win the competition.
"They were square pegs in round holes but they gave me everything they have got," he said. "There is no doubt they (Liverpool) have ability in abundance and have done exceptionally well for Rafa.
"It was an awesome team they put out. Clearly they have done a great job for the manager."