The Scottish side will face the Serie A leaders in the last 16 after Thursday's draw in Nyon having already upset the odds by emerging from a testing group featuring Barcelona, Benfica and Spartak Moscow.
Neil Lennon's men enjoyed an unforgettable victory over Barca at Parkhead having run the Spanish giants close in Camp Nou, with Lawwell admitting that they will need to emulate those displays if they are to overcome Juventus.
Lawwell told Sky Sports: "We believe we are one of the top European clubs in everything we do. Celtic should be at this level, not just for the revenue but in terms of the profile and exposure for the club.
"To get back in after two or three years has been magnificent. We go into this game with a degree of confidence. We will be ready. We have done magnificently well and I'm sure we will give a great account of ourselves.
"We have beaten the best both at home and away. It is a formidable task but we go into the game with great confidence and we will see what happens.
"We have shown we can do it, we have had some great results. It will need that level of performance, but who is to say we can't do that.
"We are delighted to be flying the flag for Scottish football, which has taken a bit of a battering recently."
Lawwell was quizzed on Celtic's transfer plans for January and admitted there could be some changes to Lennon's squad, although it is not their preferred time to do business.
He added: "Every transfer window we get to we aim to be stronger at the other end. January is not the best time to be doing it, but our intention will be to come out of the window stronger, and if that means finding some value in the market then that is what we will do."