The Hoops qualified for the knockout stages of the Champions League last week with a dramatic 2-1 home win over Spartak Moscow then followed that up with a comprehensive 3-1 victory Kilmarnock at Rugby Park on Saturday to take them three points clear of Inverness at the top of the Clydesdale Bank Premier League with a game in hand.
The Red Lichties snatched an unlikely 1-1 draw at Celtic Park in the first game with a late goal from Steven Doris, but the Parkhead boss wants his side to take care of business at the second time of asking.
"We are taking this game very seriously," he said. "It is a different competition but one where we have had no recent success and we would like to progress so it is another important game.
"The way things have been going in the last week to 10 days, you want to keep that feel-good factor going and this is potentially very dangerous.
"I have been on the end of a few cup upsets as a player and a manager and I don't want that to happen tonight."
Lennon, who has defender Mikael Lustig and midfielder Paddy McCourt back after recovering from hamstring injuries, knows his players will have to cope with the more basic surroundings of Gayfield Park, despite playing at some famous stadiums this season such as the Nou Camp in Barcelona, Spartak Moscow's Luzhniki Stadium and Benfica's Stadium of Light in Lisbon.
The former Celtic skipper said: "It will be a new experience for me going to Arbroath and for most of the players as well.
"But in the last couple of years we have been at Berwick and Peterhead (in the cup) so we know the attitude has to be right.
"I don't know what the conditions will be like so we will have to adapt to them as best we can and play accordingly."