The Hoops go into Sunday's semi-final at Hampden as firm favourites as Lennon's men chase a domestic treble.
But memories remain fresh of defeats in the last two finals of the competition for the Celtic manager - who has yet to lose against St Mirren since his appointment.
Lennon said: "People make big issues of records. The longer it goes on they talk about it more until it's broken. It can go any time.
"We are very wary of St Mirren, and with it being a semi-final and neutral venue. We're wary that our dream of winning the treble could evaporate in one game. And we need to be very careful.
"We have been to last two finals and come up short so hopefully it will be third time lucky. But we have to play well, strong and adapt to conditions as quickly as possible.
"It won't be the same as SPL game. It's a different environment and different a ball, so we need to adapt very quickly."
Meanwhile, the Celtic manager has indicated he could add to his squad before Thursday's transfer deadline after confirming Mo Bangura had joined Swedish club Elsborg until the end of the season.
Lennon added: "It's gone a bit quiet. January is a very difficult month. In my experience things hot up in last three or four days. So we are prepared for the eventuality of bringing one or two in."