The Paisley men have been handed an away trip to Queen of the South following Wednesday's second-round draw at Hampden.
Saints lifted the cup for the first time in the club's 136-year history back in March when they beat Hearts 3-2 in the final.
And Lennon insists that his 33/1 outsiders can cause another upset this year.
He said: "We had a difficult run on our way to winning the trophy last season.
"But it's no different going into this year's competition as the holders than to any other season. We want to do well in all three competitions that we enter.
"A lot of people say that lightening won't strike twice, but it can. The trophy is something we want to hold on to."
But to have any chance of retaining the trophy, the Buddies will have to do better against Jim McIntyre's Doonhamers than they managed in a 4-2 pre-season defeat at Palmerston last month.
Lennon added: "It's a tricky one for us. We had an opportunity during pre-season to go down there and play Queens.
"The positive is that we have had a wee flavour of their new astro pitch. It wasn't a pleasant night for us but Jimmy has inherited a very good squad and is now putting his own mark on it. So we know the challenge that faces us down there."
The pick of the round was the tie that will see Falkirk host local rivals Dunfermline on either Tuesday, August 27, or the following day.
Administration-hit Hearts, meanwhile, must overcome Championship side Raith Rovers at Stark's Park if they are to hope for another run to the final.
Morton will welcome Montrose to Greenock while Ross County face a 266-mile journey for their clash with third-tier opponents Stranraer.
Forfar's reward for knocking out Rangers in the first round is an away tie to Dundee, while the Dark Blues' city rivals Dundee United will face Dumbarton at the Bet Butler Stadium.
Kilmarnock host Hamilton at Rugby Park while Accies' Lanarkshire rivals Airdrie host Livingston.
Scottish Premiership new boys Partick Thistle take on Cowdenbeath in Glasgow, with Aberdeen staging their tie against Alloa at Pittodrie.